Habit Formation, the Cross Section of Stock Returns

and the Cash-Flow Risk Puzzle∗

Tano Santos

Columbia University, CEPR and NBER

Pietro Veronesi

University of Chicago, CEPR and NBER

April 6, 2010

Abstract

Non-linear external habit persistence models, which feature prominently in the recent “equity

premium” asset pricing and macroeconomics literature, generate counterfactual predictions

in the cross-section of stock returns. In particular, we show that in the absence of cross-

sectional heterogeneity in firms’ cash-flow risk, these models produce a “growth premium,”

that is, stocks with high price-to-fundamental ratios command a higher premium than stocks

with low price-to-fundamental ratios. This implication is at odds with the well-established

empirical observation of a “value premium” in the cross-section of stock returns. Substantial

heterogeneity in firms’ cash-flow risk yields both a value premium as well as most of the

stylized facts about the cross-section of stock returns, but it generates a “cash-flow risk puzzle”:

Quantitatively, value stocks have to have “too much” cash-flow risk compared to the data to

generate empirically plausible value premiums.∗We thank seminar participants at Carnegie Mellon, UCLA, Princeton University, The Federal Reserve

Bank of New York, Columbia Business School, London Business School, London School of Economics, and the

Graduate School of Business of the University of Chicago for their comments and Gene Fama, Lars Hansen, John

Heaton, and Wei Jiang for valuable suggestions. Errors, of course, remain our own. This paper has circulated

previously under the title “Cash-flow Risk, Discount Risk and the Value Premium”

1 Introduction

The equity premium and the value premium puzzles constitute two of the focal points of the

asset pricing literature. As is well-known, the starting point of the first is the inability of

standard consumption models to rationalize the observed level of the equity premium, the

volatility and predictability of returns, and the low and stable risk-free rate (see Panels A and

B of Table 1.) The value premium puzzle is instead concerned with the failure of the Capital

Asset Pricing Model (CAPM) to explain the cross-section of average returns of portfolios

sorted according to book-to-market (see Panel C of Table 1 and Fig. 1.) Surprisingly, these

two puzzles are, for the most part, studied separately. This is unfortunate because, as we argue

here, the two puzzles cannot be tackled independently: Any economic mechanism proposed to

address one of them immediately has general equilibrium implications for the other.

In this paper, we focus on one important mechanism, habit persistence, which has fea-

tured prominently both in the asset pricing literature1 and in the real business cycle literature.2

In particular, we investigate a non-linear external habit formation model a la Campbell and

Cochrane (1999), a framework particularly successful in addressing the equity premium puzzles

described above, and investigate the implications of this model for the established facts in the

cross-section of stock returns. We show that these implications are problematic and that for

this reason, the success of the non-linear habit formation mechanism has to be put on hold.

In particular, we show that when, importantly as we shall see, firms differ only in their

expected dividend growth, habit persistence models counterfactually generate a “growth pre-

mium” rather than a “value premium,” a point also recently made by Lettau and Wachter

(2007). The reason is at the heart of the habit formation model: The variation over time

of the market price of consumption risk, which is responsible for the success of the model to

explain the properties of the market portfolio, interacts with the timing of cash-flows to gen-

erate a term premium. Indeed, assets that pay far in the future are more sensitive to shocks1Sundaresan (1989), Constantinides (1990), Abel (1990), Ferson and Constantinides (1991), Detemple and

Zapatero (1991), Daniel and Marshall (1997), Campbell and Cochrane (1999), Heaton (1993,1995), Li (2001),

and Wachter (2000). All these papers focus exclusively on the time-series properties of the market portfolio and

do not investigate implications for the risk and return properties of individual securities.2For example Boldrin, Christiano, and Fisher (2001) use the habit persistence model of Constantinides (1990)

to investigate whether this mechanism can be consistent with both key asset pricing and real business cycle facts.

See also, among others, Boivin and Giannoni (2005), Christiano, Eichenbaum, and Evans (2005), Giannoni and

Woodford (2004), Ravn, Schmitt-Grohe, and Uribe (2006), and Smets and Wouters (2003,2004).

1

in the stochastic discount factor than assets with front-loaded cash-flows. For this reason the

former are riskier and command a premium over the latter. We show that “growth stocks” are

precisely those that pay far in the future and thus, they command a counterfactual premium

over value stocks.

In order to solve this “growth premium puzzle” induced by habit formation preferences,

we introduce ex ante heterogeneity in firms’ cash-flow risk, that is, firms differ from each other

not only in their expected dividend growth, but also in the covariance of their cash flows with

consumption growth itself. In this case, we find that the standard sorting procedure used

in the literature to allocate stocks to portfolios according to their price-to-fundamental ratio

endogenously selects as value stocks those with higher cash-flow risk, an implication empirically

confirmed by a series of recent papers.3 This higher cash-flow risk of value stocks naturally

translates into higher expected returns, as investors require a premium to hold stocks whose

cash flows fall at the same time as the aggregate consumption. Using simulations, we show

that if the heterogeneity in firms’ cash-flow risk is sufficiently large, then indeed stocks with

low price-dividend ratios, value stocks, do command a substantial premium compared to high

price-dividend ratio stocks.4 That is, a value premium appears. We show that, under these

conditions, the model then not only matches the properties of the market portfolio, as Campbell

and Cochrane (1999) find, but it is actually able to replicate most, if not all, of the stylized

facts about the cross-section of stock returns, including (a) the failure of the CAPM and thus,

the value premium puzzle, (b) the better performance of the conditional CAPM, and (c) the

better pricing ability of the High-Minus-Low (HML) factor as in Fama and French (1993). In

addition, the model also yields a large variation of the value premium over the business cycle,

an additional stylized fact that is well-documented in the data.5

Obviously the remaining question is then a quantitative one: is the cash-flow risk required

to match the cross-section of stock returns consistent with the data? Unfortunately not. The3See Cohen, Polk, and Vuolteenaho (2009), Campbell, Polk, and Vuolteenaho (2005), Bansal, Dittmar, and

Lundblad (2005), Parker and Julliard (2005), and Hansen, Heaton, and Li (2008). Also Liew and Vassalou

(2000) and Vassalou (2003) show that news about forecasts of Gross Domestic Product (GDP) growth correlate

with value stock returns.4 In our model, “book value” is not well defined and so we use price-dividend ratios in lieu of market-to-book

ratios throughout (see Santos and Veronesi, 2006; and Lettau and Wachter, 2007). Fama and French (1996,

Table II), Fama and French (1998, Table III), and Lettau and Wachter (2007, Table I) show that sorting by

earnings-to-price or cash-flow to price generates as sizable a “‘value” premium as sorting by book-to-market.5See for example Lettau and Ludvigson (2001) and Cohen, Polk, and Vuolteenaho (2003).

2

external habit persistence model investigated here succeeds in reproducing the times series

and cross-sectional properties of asset data at the expense of an implausible high (low) level of

cash-flow risk for value (growth) stocks when compared to the data. That is, habit preferences

induce a “cash-flow risk puzzle.” The intuition is relatively simple: Since habit preferences

tend to generate a growth premium when stocks do not differ in cash-flow risk, a large cross-

sectional difference in cash-flow risk is needed in order to “undo” the growth premium. An

extensive simulation exercise highlights the severity of the problem.

Outline of the paper. The article develops as follows. Section 2 introduces a general

equilibrium model with multiple securities which is solved for prices and returns in Section

3. This model generalizes the setting put forward in Menzly, Santos, and Veronesi (2004),

MSV henceforth, in order to have more flexible preferences and a more manageable process

for firms’ cash flows while retaining the closed-form solutions that are such an important

analytical advantage when dealing with multiple securities. Section 4 investigates qualitatively

the implications of the model for the cross-section of stock returns whereas Section 5 does the

same quantitatively. It is in Section 5.3 where we introduce the cash-flow risk puzzle. In

Section 6 we use our model to shed new light on standard asset pricing models tests. In

particular, we use our model to construct an HML factor as in Fama and French (1993) and

provide an economic foundation for its success as a cross-sectional predictor. In this section,

we also show that the model is able to match to a surprising degree the time-series variation

in the value premium, which is at the heart of the recent interest in the conditional CAPM.

Section 7 concludes.

Related literature. Our work touches on several recent papers in the literature on the cross-

section of stock returns, but differs from them in several respects. First, Santos and Veronesi

(2006) and Lettau and Wachter (2007) investigate the effect that cross-sectional differences in

cash-flow duration (as defined by the expected dividend growth) have on the cross-section of

expected returns. They both find that assets with low duration have high expected returns

and low price-dividend ratios whereas the opposite is true for high duration assets, that is the

value premium. Our work departs from these two papers in two crucial aspects.

First, both papers make assumptions to avoid the natural growth premium that comes

with the variation in the discount rate, which is necessary to match the properties of the

market portfolio: Santos and Veronesi (2006) fail to match the volatility of the aggregate stock

3

returns and Lettau and Wachter (2007) assume away general equilibrium restrictions, and

instead assume that the variation in the discount factor is unpriced by market participants.

These assumptions ensure that differences in durations generate a value premium. Duration

effects are also present in this paper, but the presence of the strong discount effects implied by

the Campbell and Cochrane (1999) model make, as explained above, cross-sectional differences

in duration generate a growth premium rather than a value premium. Santos and Veronesi

(2006) don’t have discount effects and Lettau and Wachter (2007) simply assume that they go

unpriced.6 Here, we don’t take this stand, but rather, and reasonably in our view, assume that

discount effects are both present and priced and some other ingredient is needed to generate

the value premium. Cross sectional differences in cash-flow risk are such an ingredient. Second,

the combination of cross-sectional differences in cash-flow risk and discount effects generates

the empirically documented time-series variation in the value premium, a regularity for which,

to the best of our knowledge, there is no extant theoretical explanation. Neither Santos and

Veronesi (2006) nor Lettau and Wachter (2007) address this issue.

Our paper also touches on the recent literature emphasizing cross-sectional differences in

long-run risk across asset classes. For instance, Parker and Julliard (2005) and Bansal, Dittmar,

and Lundblad (2005) use cross-sectional differences in the long-run covariance between returns,

consumption growth, and dividend growth, to offer a characterization of cross-sectional differ-

ences in one-period returns. Clearly, the long-run components of cash-flow risk are but one

contribution to one-period returns; transient components may also be very important. Rec-

ognizing this, Hansen, Heaton, and Li (2008) offer a characterization of the long-run trade-off

between risk and return. This long-run trade-off is key because transient components, which

may be first-order for one-period returns, are negligible in the long-run.7 In contrast, our

definition of cash-flow risk is entirely unrelated to low-frequency components in consumption

growth, which are (mostly) absent from our paper, and rather it emphasizes contemporaneous

covariances of consumption and dividend growth. More importantly, the discount effects that6This is the fundamental reason why duration effects are enough to generate a value premium in these two

papers. All assets have identical, and positive, cash-flow risk, but some have their dividends more front-loaded

than others. This has two consequences. First, and by assumption, the more front-loaded the dividends, the

lower expected dividend growth and thus, the lower the price-dividend ratio. Second, the more front-loaded

the dividend, the riskier the asset as it would constitute a larger fraction of current consumption and thus, the

higher the premium. Thus, these assets are “value” and a value premium arises in these two papers.7More precisely, for these authors, value stocks are riskier because their cash-flow growth process loads

relatively more than growth stocks on low-frequency components of consumption growth.

4

play so prominent a role in our calibration are entirely absent in this literature so that, once

again, the time-series variation of the value premium cannot be generated in the context of

these models. Finally, none of these papers offers an integrated view of the time-series and

the cross-section of stock returns but try to relate the properties of the market portfolio with

the cross-sectional properties of portfolios that add up to the market itself. We argue here, in

contrast with the cash-flow literature, that these two sides of the asset pricing puzzles have to

be jointly considered, otherwise the inferences on cash-flow parameters are misleading.8

Campbell and Vuolteenaho (2004) decompose shocks to market returns into shocks to

expected discount rates and shocks to expected dividend growth rates. They show that value

and growth load on these shocks differently and this, combined with the market price of

risk associated with these shocks, generates a value premium and its corresponding puzzle.

Differently from us, however, they neither connect the time-series properties of the market

portfolio with the magnitudes of the cash-flow risk needed to generate a value premium, nor

do they address the time-series variation of the value premium.

Our paper also relates to the literature on conditional asset pricing. For instance, Lettau

and Ludvigson (2001) show that empirically a conditional version of the CAPM outperforms

its unconditional counterpart. Their results provide empirical evidence supporting our model’s

implication that conditioning variables capture the time-series variation in the value premium.

In our setup, as in the data, the conditional CAPM performs better than the unconditional

CAPM. Importantly, though, in our model the conditional CAPM is a misspecified asset pricing

model, and so, with enough data, it can also be rejected.

The present paper is obviously related to MSV, but there are also many differences

with that paper. First, our model is more general than the one in MSV and the additional

flexibility is instrumental in the empirical performance of the model. In particular, while MSV

only consider the log-utility case and have approximate formulas for the cross-section of prices,

in this paper we solve for the power utility case and obtain exact solutions. Second, whereas

MSV are concerned with the time-series predictability of industry portfolios, the present paper

focuses on the cross-sectional predictability of value-sorted portfolios. This focus allows us

also to shed light on the vast literature on cross-sectional predictability, something MSV did8See also Brennan, Wang, and Xia (2004) and Brennan and Xia (2006) for a partial equilibrium model that

ties the time-series to the cross-section of stock returns. An investment-based general equilibrium model of

the cross-section is also put forward by Gomes, Kogan, and Zhang (2003) who build on the partial equilibrium

model of Berk, Green, and Naik (1999). See also Zhang (2005).

5

not touch upon and, in particular, the present paper is after a quantitative assessment of the

cash-flow risk effects needed to generate a plausible value premium.

2 The model

We consider an endowment economy with n financial assets. Each asset has an instanta-

neous dividend stream denoted by Dit, for i = 1, .., n. The consumption good is immediately

perishable and non-storable, which yields the equilibrium restriction

Ct =n∑

i=1

Dit. (1)

This add-up, general equilibrium restriction is important in our setting, as it breaks the

theoretical validity of the CAPM, as discussed below.9 Unfortunately, even relatively simple

processes for Dit imply aggregate consumption processes that are difficult to work with and

restrictive assumptions need to be made for tractability (see discussion in MSV; Santos and

Veronesi, 2006; Cochrane, Longstaff, and Santa-Clara, 2008; and Martin, 2008). We follow

MSV and Santos and Veronesi (2006), and make assumptions about aggregate consumption

Ct, and the joint dynamics of the shares of aggregate consumption produced by each asset,

denoted by

sit =

Dit

Ct. (2)

Assumption 1. Aggregate consumption is given by

dCt

Ct= µc (st) dt + σ′c dBt,

where Bt is an n× 1 vector of Brownian motions, and

µc (st) = µc + µc,1 (st) and µc,1 (st) = s′t θCF . (3)

Above, st =(s1t , ..., s

nt

)′, θCF =(θ1

CF , ..., θnCF

)′, and σc = (σc, 0, ..., 0)′ . The specification of

θiCF is explained below.

As in Campbell and Cochrane (1999), we assume consumption growth has constant

volatility. Unlike them, however, we assume expected consumption growth has a predictable9Several recent articles have emphasized the importance of market clearing conditions in finance, such as

Santos and Veronesi (2006), Johnson (2006), Cochrane, Longstaff, and Santa-Clara (2008), and Martin (2008).

None of these papers combines habit formation with multiple trees, nor investigates the properties of the cross-

section of stock returns, except for Santos and Veronesi (2006), already discussed in the introduction.

6

component that depends on the distribution of shares. We make this assumption for four

reasons: First, it follows naturally from the general equilibrium restriction (1) in any model

that has dividend processes as primitives (see Eq. (29) in the Appendix A).10 Second, this

assumption is consistent with the recent long-run risk literature, which shows a small persistent

predictable component in consumption growth (see, e.g., Bansal and Yaron, 2004; Hansen,

Heaton, and Li, 2008). In our model, such a predictable component is also small.11 Third, the

specific assumption (3) allows us to obtain analytical formulas for asset prices, an important

property given our focus on the cross-section of stock returns with many assets. Finally, in

our model the time variation of expected consumption growth breaks the theoretical validity

of the CAPM, both conditionally and unconditionally, a property that we exploit to provide

insights on the economic meaning of the tests of the CAPM provided in the literature.

Assumption 2. For each i, the share sit follows the mean reverting process

dsit = φ

(si − si

t

)dt + si

t σi (st)′ dBt, (4)

whereσi (st) = νi −

n∑

j=1

sjtνj . (5)

The cash-flow model (4) imposes a structure on the relative size of firms, where “size” is

measured as the fraction of total output produced by a given firm. In particular, it imposes the

economically plausible assumption that no firm will take over the economy, as sit > 0 for all i.

In addition, the volatility σi (st) in (5) ensures that∑n

i=1 sit = 1 for all t, which in turn implies

that (1) is always satisfied. Although the form of the volatility σi (st) in (5) seems ad hoc,

it can actually be recovered from first principles in a model with multiple dividend processes

each with constant volatility (see Appendix A). Instead, the key simplifying assumption is the

mean-reversion component in the drift rate of (4).

2.1 Expected cash-flow growth and cash-flow risk10Such models also imply that the volatility of consumption is a weighted average of dividend volatilities,

which we instead approximate to a constant.11 We show in simulations below that expected consumption growth fluctuates between a maximum of 2.22%

and a minimum of 1.87%, a very mild variation compared to the 1.5% standard deviation of consumption growth

that we assume. Indeed, for our baseline case, which has the maximum variation in expected consumption

growth, when we regress future consumption growth on ln (Pt/Ct) in artificial data we find R2s that are puny,

between 0.1% and 0.2% at the three- and four-year horizons, respectively.

7

Given Assumptions 1 and 2, we can apply Ito’s Lemma to Dit = si

tCt and obtain:

dDit

Dit

= µiD,tdt + σi

D (st)′ dBt, (6)

where the dividend drift and volatility are given by

µiD,t = µc + θi

CF + φ

(si

sit

− 1)

(7)

σiD (st) = σc + σi (st) . (8)

In these formulas,

θiCF = ν ′i · σc.

Two comments are in order: First, Eq. (7) shows that when the asset’s relative share,

si/sit, is high and thus, the asset’s relative contribution to total consumption is below its

long-term average, the asset has an expected dividend growth higher than the unconditional

expected consumption growth µc (adjusted for a small Ito term θiCF ).12 In addition, the drift

rate of dividends µiD depends on a parameter θi

CF , which is asset specific and it depends on the

correlation of the stock’s share sit with consumption growth, as shown below. While technically

θiCF is simply an Ito term obtained from the definition Di

t = sitCt, we note that quantitatively

it has a minimal impact on the average dividend growth itself: As we show in the calibration

section, θiCF is an order of magnitude smaller than the other two drift components.13

Second, in our model the stochastic discount factor is only driven by shocks to consump-

tion growth. Thus, cash-flow risk is measured by the following covariance

σiCF,t ≡ Covt

(dDi

t

Dit

,dCt

Ct

)= σ′cσc + θi

CF − s′t θCF . (9)

The conditional cash-flow risk of asset i, σiCF,t, will play a prominent role in this paper. The

term θiCF − s′t · θCF is parametrically indeterminate, that is, adding a constant to all θi

CF

leaves this term unaffected, as∑n

i=1 sit = 1. Thus, we can impose the identifiability restriction

n∑

j=1

sjθjCF = 0. (10)

12MSV find strong empirical support for the inverse relation between relative share and dividend growth in

industry portfolios.13In our model, θi

CF s are uniformly distributed around the interval [−θCF , θCF ]. The maximum level of

θCF = 0.0035, which is much smaller than both the assumed average consumption growth µc = 2% and the

fluctuations in expected dividend growth induced by the third term in (7), φ(si/sit − 1), which is over 10%.

8

The expected covariance between asset i’s cash-flow growth and consumption growth is

σiCF = E

[σi

CF,t

]= E

[Covt

(dDi

t

Dit

,dCt

Ct

)]= σ′cσc + θi

CF . (11)

The parameter θiCF then regulates the relative cash-flow risk of individual assets. Notice that

the benchmark level of risk of an asset is the riskiness of aggregate consumption: An asset

is risky (safe) if its cash-flows are more (less) risky than aggregate consumption. This is a

general equilibrium restriction as, by definition, the variance of consumption growth must be

a weighted average of its covariances with individual dividend growth. Throughout we refer

to either σiCF or θi

CF as “cash-flow risk” as there is a one-to-one mapping between them.

We conclude this section by emphasizing that the present framework can be generalized

to introduce more realistic features but, clearly, at the cost of additional complexity. For

instance, we assume for simplicity that firms are infinitely lived and that agents know the

long-term average size si. A plausible generalization is one in which si is unknown, and agents

learn about it over time as they observe different dividend and consumption realizations. In

this case, the pricing function, Eq. (20) below, will depend on the expected long-term share

Et[si] rather than on si. This extension though would largely leave the results unaffected.

Indeed, standard filtering results imply that the variation in Et[si] would be independent of

consumption shocks, as consumption does not yield any additional information on si that is

not already in the shares themselves (see also Pastor and Veronesi, 2003). It follows that this

additional variation would be unpriced and thus, would have no impact on firms’ expected

returns. Second, since expectations move more slowly than signals, the ratio Et[si]/sit would

still tend to move inversely with sit, exactly as in the case in which si is known. Since the

ratio si/sit, as is formally shown in Propositions 2 and 3, is the key variable affecting the

firm’s expected dividend growth, its price/dividend ratio, and its expected return, it follows

that the cross-sectional relation between price/dividend ratios and expected returns would not

change if we were to assume that si was unknown. Finally, the learning model also partially

addresses our assumption of an infinitely lived firm: assuming si are randomly selected at time

zero, some firms would then converge to very low “sizes,” effectively disappearing from the

economy.14 In summary then, adding learning to the model to account for the fact that the

agents are not likely to know the long-run contributions of the different firms to the overall14We also solved the model assuming exponential distributed exit times (firm death), which lead to the usual

increase in the time discount. The results remain the same, but the model becomes more challenging as to keep

it stationary, we must have a flow of firms entering the economy.

9

economy substantially complicates the analysis without largely affecting the relation between

price/dividend ratios and expected returns, which is the focus of this paper.

2.2 Preferences and Habit Dynamics

There is a representative investor who maximizes

E

[∫ ∞

0u (Ct, Xt, t) dt

], (12)

where the instantaneous utility function is given by

u (Ct, Xt, t) = e−ρt (Ct −Xt)1−γ

1− γ. (13)

In Eq. (13), the variable Xt denotes an external habit level and ρ denotes the subjective

discount rate. In Campbell and Cochrane (1999) the fundamental state variable driving the

attitudes towards risk is the surplus-consumption ratio, St = (Ct −Xt) C−1t . To obtain closed-

form solutions for prices when there are multiple securities, MSV use a log habit model and

specify instead the inverse surplus S−1t as a mean-reverting process. MSV’s modeling device

though cannot be applied when γ > 1 and, moreover, they only obtain approximate formulas

for the case θiCF 6= 0. The present paper offers a generalization of MSV that allows us to

handle a large class of models. The key ingredient in this generalization is to focus on the

process

Gt =(

Ct

Ct −Xt

)γ

= S−γt . (14)

To obtain a plausible, yet tractable, model for the dynamics of Gt, consider first the

implications for Gt under the standard assumption that Xt is an exponentially weighted average

of past consumption levels, as in Constantinides (1990) and Detemple and Zapatero (1991),

Xt = λ∫ t−∞ e−λ(t−τ)Cτdτ. An application of Ito’s Lemma to (14) yields the process

dGt =[µG (Gt)− σG (Gt) µc,1 (st)

]dt− σG (Gt) σcdB1

t , (15)

where µG (Gt) and σG (Gt) > 0 are complicated functions of Gt, provided in Eq. (31) and (32)

in Appendix A. Eq. (15) shows that a higher expected consumption growth µc,1 (st) implies

a lower drift rate of Gt. Intuitively, an increase in the expected growth rate of consumption

implies a high future level of consumption relative to the current habit Xt and thus, a higher

surplus consumption ratio St, and, given (14), a lower expected Gt. As in MSV and Campbell

and Cochrane (1999), we make specific assumptions on µG (Gt) and σG (Gt) in (15) to obtain

a more manageable process. In particular, we assume

µG (Gt) = k(G−Gt

)and σG (Gt) = α (Gt − λ) . (16)

10

The first component of the drift of Gt is a mean-reversion component and captures the

basic idea of habit persistence models, namely that the habit Xt eventually “catches up” with

Ct. The second component, as discussed above, links the drift rate of Gt to µc,1 (st). As for the

diffusion component, and as in MSV, λ ≥ 1 bounds Gt from below at λ and α > 0 transmits

the innovations in consumption growth, dB1t , to the convexity of the utility function.15 Note

that MSV’s model is a special case of (15) and (16) and obtains when γ = 1 and consumption

growth is i.i.d., which is achieved by setting µc,1 (st) = 0.

3 Equilibrium asset prices and returns

3.1 The total wealth portfolio

The next proposition generalizes the results in MSV to the present model in what con-

cerns the total wealth portfolio, that is, the claim to the aggregate consumption process.

Proposition 1. The price-consumption ratio, the expected excess return, and diffusion terms

of the total wealth portfolio are, respectively:

P TWt

Ct= αTW

0 (st) + αTW1 (st) Sγ

t (17)

Et

[dRTW

t

]= (γ + α (1− λSγ

t ))

Sγt α (1− λSγ

t )fTW1 (st) + Sγ

t

σ2c +

n∑

j=1

wTWjt σj

CF,t

(18)

σTWR,t =

Sγt α (1− λSγ

t )fTW1 (st) + Sγ

t

σc +n∑

j=1

wTWjt σj

D (st) , (19)

where αTW0 (st), αTW

1 (st), fTW1 (st) and

wTW

jt

are given in Appendix B.

As in Campbell and Cochrane (1999) and MSV, the price-consumption ratio of the total

wealth portfolio is increasing in the surplus-consumption ratio St: A high St implies a low

local curvature of the utility function, a “less risk-averse” attitude of the representative agent,

and thus, a higher price-consumption ratio. Unlike Campbell and Cochrane (1999) and MSV,

the price-consumption ratio now depends on the entire vector of shares st. Intuitively, this

effect stems from our assumption about consumption growth predictability (see the discussion

after Assumption 1). The functions αTW0 (st) and αTW

1 (st) are typically decreasing in expected15Clearly, the assumptions (16) then imply that habit Xt is no longer the weighted average of past consump-

tion, as above, but a more complicated non-linear function of past consumption shocks. See Campbell and

Cochrane (1999) for a discussion. See also Hansen (2008) for a discussion of the risk-return implications of our

habit model proposed above.

11

consumption growth because in our setup, the elasticity of intertemporal substitution is less

than one. Thus, this component implies that an increase in µc (st) results in lower prices.16

As for the expected excess returns, (18), and the volatility of returns, (19), we postpone the

discussion of the intuition of these expressions until Section 5.2, when we assess quantitatively

the implications of the model.

3.2 Prices and returns for individual securities

The next proposition delivers closed-form solutions for individual stock prices:

Proposition 2. The price of asset i is given by

P it

Dit

= αi0 + αi

1Sγt + αi

2 (st)(

si

sit

)+ αi

3 (st)Sγt

(si

sit

), (20)

where αi0, αi

1 are positive constants and αi2 (st) and αi

3 (st) are positive linear functions of the

share vector st given in Appendix B.

As before, a higher surplus-consumption ratio St, which implies lower “risk aversion,”

or a higher expected dividend growth, as measured by the relative share si/sit (see (7)), result

naturally in higher price-dividend ratios. The last term in (20) shows that shocks to the

surplus-consumption ratio have a stronger effect on the price-dividend ratio the higher the

asset’s expected dividend growth. This is linked to the duration effect that plays so prominent

a role in what follows. Finally, as it was true for the total wealth portfolio and for the same

reason, the price of each individual asset also depends on functions of the vectors of shares

αi2 (st) and αi

3 (st).

The next proposition presents a characterization of expected excess returns. The intu-

ition and implications of Proposition 3 are given in depth in Section 4.

Proposition 3. The expected excess return of asset i is given by

Et

[dRi

t

]= µDISC

i,t + µCFi,t ,

16To review the economic reasoning, a low elasticity of intertemporal substitution implies a taste for consump-

tion smoothing. An increase in expected consumption growth yields a higher desire for current consumption, and

thus, lower savings. Because stocks and bonds are less desirable now for the representative consumer, prices have

to drop in order to encourage him to hold them, resulting, for example, in a decrease of the price-consumption

ratio of the total wealth portfolio.

12

where

µDISCi,t = (γ + α (1− λSγ

t ))

Sγ

t

f i1

(si

sit, st

)+ Sγ

t

α (1− λSγ

t ) σ2c (21)

µCFi,t = (γ + α (1− λSγ

t ))

1

1 + f i2 (St, st)

(si

sit

) + ηiit

σi

CF,t +∑

j 6=i

ηijtσ

jCF,t

, (22)

with

f i1

(si/si

t, st

)=

αi0 + αi

2 (st)(si/si

t

)

αi1 + αi

3 (st)(si/si

t

) > 0 and f i2 (St, st) =

αi2 (st) + αi

3 (st) Sγt

αi0 + αi

1Sγt

> 0,

and ηijt is given in expression (39) in Appendix B.

4 Growth versus value premiums

The key empirical observation in the cross-sectional literature is that growth assets, which are

those with high prices relative to fundamentals, say price-dividend ratios, have on average lower

returns than assets with low price-dividend ratios, value stocks. In this section we investigate

what is required of the model to generate qualitatively this fact. For this we make use of the

results in both Propositions 2 and 3 above.

4.1 Discount risk effects and the growth premium

We start by focusing on the component of the premiums µDISCi,t in (21), which is the

part of the premium that is driven by variation of the aggregate discount−proxied by Sγt . To

interpret this term further, notice first that

∂P it /P i

t

∂Sγt /Sγ

t

=Sγ

t

f i1

(si/si

t, st

)+ Sγ

t

(23)

is the elasticity of prices to shocks in the variable driving the aggregate discount, which is Sγt .

The volatility of these discount shocks is

α (1− λSγt ) σc, (24)

which is the diffusion component of dSγt /Sγ

t , the inverse of our state variable Gt, as it follows

from a basic application of Ito’s Lemma to (15) . Clearly, only the component of these shocks

that covaries with the shocks to the stochastic discount factor is priced. From Eq. (33) in

Appendix B, the diffusion term of the habit stochastic discount factor is

σm = − [γ + α (1− λSγt )]σc. (25)

13

The component of the asset’s premium that is linked to discount effects is then the

product of (23), (24), and (25), which is expression (21) in Proposition 3.

Cross-sectional variation in the discount effects can only be driven by differences in the

price elasticity (23), which is in turn driven by the behavior of the function f1

(si/si

t, st

). We

have been unable to obtain a general characterization of this function, but for parameter values

that are empirically relevant we find that

∂f i1

(si/si

t, st

)

∂(si/si

t

) < 0,

and thus, assets with a higher expected dividend growth, as measured by the relative share

si/sit, display stronger discount effects. The intuition is straightforward: stocks with a high ex-

pected dividend growth pay the bulk of their proceeds far in the future. Thus, minor variations

in the aggregate discount rate− through the risk aversion of the representative investor−result

in large percentage variations of the price of the asset. This variation is naturally priced and

thus, the higher required premium of assets with high relative shares.

4.1.1 The growth premium

We can now relate these findings to the observation that when only discount effects are

present, a growth premium arises. For this it is useful to turn to Fig. 2, where we plot µDISCi,t ,

as given by (21), as a function of our proxy for expected dividend growth, si/sit, for the case

in which all firms have identical cash-flow risk, that is, θiCF ≈ 0 for all i. To generate this plot,

the level of surplus St is set to its steady state value S and the parameters used are those of the

calibration exercise discussed in detail in Section 5.1. When all firms have identical cash-flow

risk, expression (20) implies that sorting assets according to their price-dividend ratio (P/D)

is akin to sorting them on expected dividend growth, si/sit. Since low price-dividend ratio

stocks are those with low relative shares si/sit, value stocks are those located on the left-hand

side of Fig. 2 and thus, have low expected excess returns. Similarly, high price-dividend ratio

stocks are those with high si/sit and thus, growth stocks are on the right-hand side of Fig. 2

and have high expected excess returns. Thus, if cross-sectional differences in cash-flow risk are

“small,”so that Et

[dRi

t

] ≈ µDISCi,t for all stocks, growth stocks command a higher premium

than value stocks, that is, a “growth premium” is obtained.

To reinforce this point, we conduct an extensive simulation, that we describe in detail

below, to reproduce the sorting procedure that is standard in the literature on the cross-section

of stock returns. Our purpose is to replicate Fig. 1, where we plot average (log) market-to-

book of value-sorted portfolios versus average excess returns. The equivalent in simulated data

14

for the case in which firms have homogenous cash-flow risk is reported in the top panel of Fig.

3. The figure clearly shows that stocks with high average price-dividend ratios yield a higher

average return, in contrast with the data in Fig. 1. To summarize then, if discount effects

were to be the only ones present, the cross-section of excess returns would display a growth

premium rather than the value premium that is observed empirically.

4.2 Cash-flow risk effects and the value premium

The source of premiums related to cash-flow shocks, µCFi,t given in (22), has two compo-

nents to it. The first is related to shocks in the asset’s dividends and the second is related to

shocks in the dividends of the rest of the assets in the economy, which, as shown in (20) , affect

the price of asset i as well. The logic for the sources of the premiums linked to cash-flow shocks

is the same as in the discount effects case. First, it can be easily shown that the elasticity of

the price with respect to shocks to its own dividends is,

∂P it /P i

t

∂Dit/Di

t

=1

1 + f i2 (St, st)

(si

sit

) + ηiit.

Recall also that we denote σiCF,t = covt

(dDi

t/Dit, dCt/Ct

)(see Eq. (9)). The first term of

µCFi,t is then the component of the dividend shocks that covaries with shocks to the stochastic

discount factor multiplied by the effect that these shocks have on the price of asset i, as

measured by the price elasticity. As for the second term in µCFi,t , it can be shown that

∂P it /P i

t

∂Djt /Dj

t

= ηijt for j 6= i.

As before, this component of the premium results from the product of this (cross) elasticity

and the priced component of the shock to asset j’s dividends, σjCF,t.

How does the current level of expected dividend growth, as measured by si/sit, affect the

cash-flow risk component of expected stock returns? Given the conditional covariance of the

dividend of asset i with aggregate consumption, σiCF,t, the first term of (22) is unambiguous:

Since f i2 (St, st) > 0, if the asset is “risky,”that is, if σi

CF,t > 0, then a high expected dividend

growth translates in a lower premium stemming from current dividend volatility. The intuition

is also clear: a stock that pays more in the future than today has a relatively low dividend

compared to the future. Thus, the risk embedded in current dividends, σiCF,t, has a relatively

low impact on the total risk of the stock. In the limit, if the stock does not pay any dividend

today, it cannot have any “cash-flow risk,”as there is zero current covariance of dividends

with consumption. If instead the asset’s dividends covary negatively with consumption growth

15

(σiCF,t < 0), then a high expected dividend growth increases the risk premium. The argument,

of course, is the converse of the previous one.

The effect that the current expected dividend growth of asset i has on the second term of

the cash-flow risk component of stock return (22) is more difficult to tell. However, we found

numerically that, on average, the cash-flow component of expected return is increasing in σiCF ,

although variation in shares sit generate small deviations from this increasing pattern.

Finally, we note that in our model, the risk premium only depends on the relative share

si/sit and not on the steady state dividend share si per se. The reason is that there are two

forces at play when considering the effect of si on required premiums. First, a stock with a

high average dividend share si is more exposed to consumption risk, on average, but second, it

also has a higher average price. This higher price implies a lower percentage sensitivity of the

stock to consumption shocks. Since risk premiums depend on percentage returns, these two

forces exactly offset each other in our model.17 The current level of dividends is instead key

in determining the current cash-flow risk, as it is the covariance between consumption growth

and current dividends that has a direct bearing on the riskiness of the stock.

4.2.1 The value premium

We showed in Section 4.1 that the sole presence of discount effects generates a coun-

terfactual growth premium. To see whether cash-flow effects can produce the desired value

premium instead, we turn to Fig. 4. The first two panels report, respectively, the discount,

µDISCi,t , and the cash-flow risk component, µCF

i,t , of expected returns. Panel C adds up both

components to obtain Et

[dRi

t

]. Let us start with Panel A, which reports the same quantity

as in Fig. 2, µDISCi,t , but for the case in which θi

CF differ across firms. Interestingly, we see

that higher cash-flow risk increases the level of the discount component of the expected return.

The reason is that a higher cash-flow risk decreases the price of the asset, on average. Thus,

shocks to the aggregate discount (Sγt ) have a larger percentage impact on the stock price, and

thus, imply a higher risk. Nonetheless, for given cash-flow risk level, θiCF , the relation between

µDISCi,t and expected dividend growth si/si

t is positive, as discussed in the previous section.

Panel B plots the cash-flow component to expected return, µCFi,t , as a function of expected

17That expected returns are independent of si is also a feature of the standard asset pricing model. Indeed,

consider the case where the representative consumer has the standard power utility function, sit = si for all t

and, finally, let Ct follow a simple geometric Brownian motion. In this case, P it = siCtK where K is a constant.

Since the risk premium is µi = γCov`dP i

t /P it , dCt/Ct

´and dP i

t /P it is independent of si, the two assets have

identical risk premiums, independently of si. Finally notice that the asset with a higher si has a higher price.

16

dividend growth, as proxied by si/sit, for various levels of θi

CF each corresponding to a line in

the plot. As explained in the previous section, the cash-flow risk component of expected excess

returns is decreasing in the expected dividend growth for stocks with high cash-flow risk.

Panel C reports the total expected return for each asset obtained by adding the cash-flow

risk component µCFi,t to the discount risk component, µDISC

i,t . Value stocks (assets with low

P/D ratio) have, on average, high risk (σiCF ) and low expected dividend growth (si/si

t). This

combination corresponds to the area around the top-left corner of the plot in Panel C, that is,

to high expected excess return. Conversely, growth stocks (assets with high P/D ratios) must

have a combination of low σiCF and high si/si

t. This combination can be found on the bottom-

right corner of the plot in Panel C, that is, low expected return. As can be seen then, value

stocks will command a high premium and growth stocks a low (and even negative) premium.

Thus, if cross-sectional differences in cash-flow risk are “large,”then value stocks have higher

expected excess returns than growth stocks and a “value premium” is obtained.

To better illustrate this point, the top panel of Fig. 5, as it was the case with Fig. 3, again

plots the average price-dividend ratios of price-dividend sorted portfolios against their average

excess returns in simulated data, but now for the case in which firms have heterogeneous cash-

flow risk. Our purpose is to assess to what extent the model can reproduce Fig. 1, which

is obtained from historical data. As in Fig. 1 and in contrast with Fig. 3, the presence

now of heterogeneity in cash-flow risk generates a value premium: Low price-dividend ratio

stocks, value stocks, are those that earn a high average excess return. The model is thus, able

to generate a value premium, although, clearly, the question is whether it can do so with a

reasonable cross-sectional dispersion of cash-flow risk.

4.3 Conditional versus unconditional value premiums

A novel theoretical implication of our framework is that the presence of discount risk

effects, which are associated with the time-series variation in risk preferences, affects the dy-

namics of the value premium, a feature for which there is already some empirical evidence

(Cohen, Polk, and Vuolteenaho, 2003, Table V). Essentially, discount risk effects interact with

the cross-sectional dispersion in cash-flow risk to induce fluctuations in the value premium,

as shown in Fig. 6. This figure plots the expected excess returns of three assets against the

surplus-consumption ratio, St. The dotted line shows the expected excess return of the market

portfolio; the solid line corresponds to the expected excess return of a representative value stock

with high cash-flow risk and low expected dividend growth; finally the dashed line corresponds

to the premium of a representative growth stock with low cash-flow risk and high expected

17

dividend growth. As can be seen, when the surplus-consumption ratio is low (high), the value

premium is high (low): Assets with a high value of θiCF are particularly riskier when the

representative agent is highly risk-averse which occurs whenever adverse consumption growth

shocks depress the surplus-consumption ratio, increasing in turn the market premium and its

dividend yield. Thus, in our model, the value premium has a strong predictable component,

being high (low) when the market premium is high (low).

5 Quantitative implications of the model

In this section, we conduct a simulation study to evaluate the extent to which the model

can match the standard return moments both in the time-series and the cross-section, which

can be found in Table 1. The empirical data set is standard and is briefly described in the

legend to Table 1. Panel A shows the mean and standard deviation for the returns on the

market portfolio and the risk-free rate. Panel B shows the predictability regressions of Fama

and French (1988) and Campbell and Shiller (1988) for two different sample periods, which are

meant to emphasize the sensitivity of these results to the particular period under consideration.

Panel C shows the standard statistics for the cross-section of book-to-market sorted (decile)

portfolios. In particular, we report average excess returns for the ten value-sorted portfolios for

two sample periods, 1948–2001 (Panel C-1) and 1926–2001 (Panel C-2). The value premium

is 5.50% in the 1948–2001 sample, which is very similar to the corresponding one in the longer

sample. For the shorter sample, we also report the average market-to-book, the Sharpe (1964)

ratio, and the price-dividend ratio, the latter being the variable along which we are going to

be sorting portfolios in simulated data as the model lacks a counterpart for the book value.

Notice a strong feature of the data: Value stocks have higher Sharpe ratios than growth stocks

and indeed, from the highest market-to-book portfolio to the lowest, the Sharpe ratio almost

doubles

An important regularity concerns the CAPM alphas and betas of these portfolios. For

the postwar sample, there is a flat if not slightly negative correlation between the CAPM betas

and average returns, which is at the heart of the value premium puzzle. Indeed, the alphas

of value stocks are positive and statistically significant and the extreme growth portfolio is

negative and also statistically significant.

The evidence is somewhat different for the prewar sample. Panel C-2 reports the annual-

ized monthly average excess returns for the ten value-sorted portfolios for a sample period going

18

back to 1926 as calculated by Ang and Chen (2007, Table 1, Panel A.) Relative to the earlier

sample, the CAPM betas correlate positively with average returns, rather than negatively, and

this gives some hope for the CAPM to address the value premium.18

5.1 Details of the simulation

We simulate 10,000 years of quarterly data for 200 firms that we then sort into ten

portfolios according to their price-dividend ratio (see footnote 4) in an effort to mimic the

standard procedure used in the cross-sectional literature. Table 2 contains the parameter values

that are used throughout. We set the average consumption growth at 2% and its standard

deviation at 1.5%, which should be measured against the value in the postwar sample of 1.22%

and the one for the longer sample starting in 1889, which is 3.32% (see Campbell and Cochrane,

1999, Table 2.) We choose γ = 1.5, which is between the values used by MSV, γ = 1, and

Campbell and Cochrane (1999), γ = 2. This choice implies a steady state value of the local

curvature of the utility function of γS−1 = 48, higher than the already high value of Campbell

and Cochrane (1999) which is 35. The minimum value of this local curvature is 27.75. Finally

the parameters k and α are similar to the values chosen by MSV.

As for the share process, all our results depend on the ratio si/sit and not the level

si, and so we set si = 1/200 = 0.005, without loss of generality. The cross-section of stock

returns is sensitive though to parametric choices of other cash-flow parameters, θCF , φ, and νi.

To avoid parameter proliferation, we restrict the share volatility νi = (νi,0, 0, ..., 0, νi,i, 0, ..).

Given a value for the cash-flow risk parameter, θiCF , the first entry by definition must be

νi,0 = θiCF /σc. The second entry−the idiosyncratic part−is chosen constant across all assets

according to the formula, ν2i,i = ν2−max(ν2

i,0), where ν is a chosen parameter. In other words,

ν is the maximum share volatility across assets.

We report first the results for our benchmark case where φ = 0.07, which is the value

that MSV (Table I) estimate for the market portfolio, and ν = 0.55, and θCF = 0.345%,

which as we will show shortly are values that allow us to match the moments reported in

Table 1. Section 5.3 contains a thorough discussion of the economic significance of these latter

assumptions, focusing especially on their impact on the main trade-off we highlight in this

paper: the tension between discount effects and cash-flow risk effects in what concerns the

cross-section of stock returns. In what follows, we refer to θCF as the cash-flow risk parameter,

but the reader should keep in mind that it is the support of the cash-flow risk parameters of18On this point, see also Campbell and Vuolteenaho (2004) and Fama and French (2006), who also show that

notwithstanding the evidence above, the CAPM is still rejected in the longer sample.

19

individual assets.

5.2 Simulation of the model

Table 3 is the counterpart to Table 1 but in simulated data.19 As can be seen the

model does a reasonable job at capturing the main patterns of the empirical sample. Panel

A shows that the model generates a sizable equity premium, though a bit low compared to

other simulations of external habit persistence models, and high volatility of stock returns. As

in MSV, the model yields a low risk-free rate though with volatility somewhat higher than

its empirical counterpart. Panel B shows that the model produces the predictability at long

horizons, though the R2s are lower than the ones observed in the empirical data.20

Why is the equity premium lower in our model than in other external habit persistence

models, such as Campbell and Cochrane (1999) and MSV? This is the result of a feature of our

model that is absent from Campbell and Cochrane (1999) and MSV, namely, the small pre-

dictable variation in expected consumption growth due to general equilibrium restrictions (see

the discussion after Assumption 1.) To gauge the intuition of this result, it is useful to return

to Proposition 1. Consider the case where there is a positive shock to consumption growth.

This immediately translates into a higher price, which is now a claim to a larger dividend.

In habit persistence models, this positive consumption shock gives a second positive jolt to

prices through the increase in the surplus-consumption ratio, S, which lowers the representa-

tive agent’s risk aversion. Because this makes stocks more volatile and riskier, they command

a larger premium. This is the standard effect in habit persistence models and it corresponds

to the first terms in (18) and (19) for the expected return and volatility, respectively.

In our framework though, shocks to consumption growth and shocks to expected con-

sumption growth are positively correlated. Thus, on average, in the presence of a positive

consumption growth shock, expected consumption growth is also high and this makes the to-

tal wealth portfolio less desirable to a representative consumer with a strong preference for

intertemporal smoothing, as is standard in habit persistence models.21 This is a negative force

on prices which partially undoes the positive effects discussed above. As a result, the volatility19We do not report t-statistics of simulation results, because the large sample (40,000 quarters) makes them

meaningless. We take the simulated values as population moments and compare them with their empirical

counterparts.20Still, the riskless rate volatility is vastly lower than the one of traditional habit persistence models, such as

Abel (1990) and Boldrin et al. (2001), who report a riskless rate volatility of 17.87% and 24.6%, respectively.21Yang (2007) proposes a model that combines the Epstein-Zin utility framework with habit persistence

precisely to allow for a more flexible specification of the intertemporal elasticity of substitution.

20

is lower and so is the required premium when compared to the standard habit model with i.i.d.

consumption growth. This is the second term in both (18) and (19). Notice that this effect

is also bound to affect the strength of the predictability regression. It is important to note,

however, that our model does not generate too much predictability of consumption growth, as

already discussed in footnote 11.

Panel C of Table 3 reports the quantitative implications of the model for the cross-

section. The model generates a sizable value premium of a bit above 5%, though the average

returns of individual portfolios are off by as much as the equity premium is. Interestingly,

Sharpe ratios in simulated data show the same increasing pattern when moving from growth

to value as in the empirical sample: Value stocks are good deals according to this metric both

in the data and in the model. The model is able to generate not just the value premium

but the value premium puzzle as well. Indeed, notice that the CAPM alphas are negative for

two growth portfolios (portfolios 1 and 2) and positive for the rest of the portfolios, which

matches surprisingly well the empirical sample in Table 1. Importantly though, the CAPM

betas covary positively with average returns in the cross-section, a pattern that is consistent

with the 1926-2001 sample (see Panel C-2 of Table 1), but not the postwar sample.

Why does the CAPM fail in our setting? As mentioned already, our model features a

mild time variation in expected consumption growth through the term µc,1 (st) = s′t θCF in (3).

The price/consumption ratio of the total wealth portfolio in (17) is a non-linear function of

both the surplus-consumption ratio St and expected consumption growth, P TWtCt

= αTW0 (st) +

αTW1 (st) Sγ

t . This result shows that general equilibrium restrictions imply that returns on

the total wealth portfolio P TW are determined by two types of shocks. First, consumption

shocks dC, which affect the price P TW through the level of consumption itself, the surplus-

consumption ratio St, and their impact on the systematic component of the variation in shares

st. The second source of variation of the total wealth portfolio P TW is the component of the

variation in expected consumption growth that is orthogonal to consumption shocks. This is

induced by the idiosyncratic components of shares st. As these vary, expected consumption

growth changes and so does the price P TW , but this variation is not priced by the habit-

based stochastic discount factor. In essence, the idiosyncratic component of the variation in

expected consumption growth breaks the perfect correlation between the total wealth portfolio

return and the habit stochastic discount factor, which in turn invalidates the CAPM, both

conditionally and unconditionally. We emphasize that, as discussed after Assumption 1, in

multi-asset models the variation in expected consumption growth results from the general

21

equilibrium restriction (1). It follows that, in general, we should not expect the CAPM to hold

in a model with multiple trees.22

In conclusion, our model is able to capture to a surprising degree the main characteristics

of the return distribution both in the time-series and the cross-section and thus, seems a useful

lens through which to draw inferences about cash-flow parameters. Specifically, is the value of

θCF needed to match these moments “high” or “low”? We turn to this question next.

5.3 The cash-flow risk puzzle

In this section, we assess what our choice of the cash-flow risk parameter θCF means for

the properties of the cash-flow process. Evaluating the assumed magnitude of the cash-flow risk

parameter is not easy, as it requires observing asset distributions that accrue to consumers, and

then calculate the hard-to-estimate correlation with consumption growth. Here, we get at this

question by measuring instead the model implied “cash-flow betas,” as defined and estimated

in empirical data by Cohen, Polk, and Vuolteenaho (2009), and compare the model-implied

values to their empirical counterparts.

Specifically, using data from 1928 to 1999, Cohen, Polk, and Vuolteenaho (2009) regress

different measures of firms’ cash-flows on the corresponding measures of market cash-flows as

in, for example,

R−1∑

j=0

ρjCPV ∆dp

t+j,j+1 = βpCF,0 + βp

CF,1

R−1∑

j=0

ρjCPV ∆dmkt

t+j + εpt+R−1 (26)

for each time t and each portfolio p = 1, ..., 10. Here, ∆dpt+j,j+1 is the dividend growth at time

t + j of the portfolio p which was formed j + 1 years earlier, that is, at t− 1. Similarly, ∆dmktt+j

is the dividend growth of the market at time t+j. Finally, ρjCPV = 0.95 is a discount, and R is

the number of years over which the average growth rate is computed. They call the regression

coefficient βpCF,1 the cash-flow beta, as it measures essentially how portfolio cash-flows covary

with aggregate cash-flows.

The empirical estimates of Cohen, Polk, and Vuolteenaho (2009) are reported in Table 4

Panel A. Notice first that empirically, irrespective of the cash-flow measure used, value stocks

have higher cash-flow betas than growth stocks, though magnitudes differ across measures. If

either (accumulated) return on equity,∑4

j=0 ρjROEpt+j,j+1, or (accumulated) dividend growth

22We expressed the issue in terms of shares, but this finding is true more generally. Indeed, in the general set

up in Appendix A we may assume vi = [vi1, 0, ..., 0, vi

i , ...]. Shocks to consumption then depend only on dB1,

the first Brownian motion, as the others are diversified away. However, expected consumption growth depends

on the shares, µ (st) =P

i sitµ

i, whose movement does depend on an aggregation of the idiosyncratic shocks.

22

is used as a measure of cash-flow growth, the regression coefficients roughly double when we

go from growth to value stocks. If instead we use (accumulated) earnings growth relative to

market value,(Xp

t+4,4 −Xpt−1,0

)/MEp

t−1,0, the coefficients increase by a factor of ten. Finally

if (accumulated) earnings relative to market,∑4

j=0 ρj(Xp

t+j,j+1/MEpt+j−1,j

), is used they

increase almost by a factor of 20.23

Turning now to the model, the first line of Table 4, Panel B reports the cash-flow betas

as estimated from the same regression (26) in simulated data. As one can see, the model

generates the striking pattern uncovered by Cohen, Polk, and Vuolteenaho (2009): Cash-flow

betas increase with book-to-market (i.e., dividend yield in our model).24 This result of our

model stems from the fact that, on average, the procedure of sorting stocks based on their

price/dividend ratio endogenously selects as value stocks those with higher cash-flow risk, as

illustrated in Section 4 (see Fig. 4). Indeed, this effect of the sorting procedure can be seen

in the second line of Panel B, which shows that the average cash-flow risk parameter θiCF is

higher for the value portfolios than for the growth portfolios. We emphasize that the fact that

value stocks have a higher cash-flow risk than growth stocks in our model is a result and not

an assumption, as it solely stems from the sorting procedure.

The cash-flow risk puzzle, which we highlight in this paper, is the observation that

even if the model qualitatively implies that cash-flow betas increase with the dividend yield,

quantitatively the estimates of the cash-flow betas in simulated data are too high, in absolute

values, relative to their empirical counterparts. Indeed, notice two facts: First, our model

produces a spread of cash-flow betas that is comparable to the empirical spread only when cash-

flows are measured as the ratio of earnings-to-market. For any other measure, the empirically

observed spread is much lower. Second, our model generates cash-flow betas for growth stocks

that are negative and large in absolute values, which is also at odds with the data, as all

empirical cash-flow betas, independently of the measure used, are positive.23Other authors have also found that value stocks have a higher cash-flow risk than growth stocks. For

instance, Bansal, Dittmar, and Lundblad (2005) regress market-to-book sorted portfolios’ dividend growth on

a moving average of consumption growth rates, and find that indeed cash-flow betas are larger for value-sorted

portfolios (see Table 1, Panel A). Similarly, Hansen, Heaton, and Li (2008) show that growth stocks have low

long-run cash-flow covariation with consumption relative to value. We focus on the Cohen et al. (2009) measure,

as it is directly related to our setting, while the measure of cash-flow risk in Bansal et al. (2005) and Hansen et

al. (2008) is related to the loading on the small predictable component in expected consumption growth.24Cohen, Polk, and Vuolteenaho (2009) argue that this cross-sectional dispersion of cash-flow betas can explain

much of the long-horizon returns of value-sorted portfolios.

23

Clearly the question is whether our choice of θCF can be relaxed in order to reproduce

the return moments of interest in the data while at the same time generating estimates of cash-

flow betas more in line with the data. Tables A.1 and A.2 in Appendix C contain summary

results of simulations under several alternative values of the cash-flow parameters (θCF , φ,

ν), which are also discussed in detail there. Fig. 7 summarizes these results. The figure

plots the cash-flow beta spread in simulated data against the corresponding value premium

for all these different parameterizations of the cash-flow processes. We also plot Cohen et al’s

(2009) estimates of these cash-flow betas (in diamonds) against the empirically observed value

premium, which is denoted by the vertical dashed line. The key to our model is that there is

a positive relation between the spread and the value premium. As we increase the spread in

cash-flow risk, as measured by θCF , we increase the cash-flow beta spread, β10CF,1 − β1

CF,1, and

the model does a better job at matching the value premium. The point of this paper is that

the cash-flow beta spread needed to get the model to the vertical line is too high relative to the

empirical measures estimated by Cohen, Polk, and Vuolteenaho (2009). The only exception is

when these authors use clean surplus earnings-to-market as their measure of cash-flows. Recall

though that even this measure implies positive cash-flow betas for growth stocks whereas our

model implies negative ones.

To summarize, a habit formation model calibrated to match the properties of the mar-

ket portfolio needs too much dispersion in cash-flow risk in order to quantitatively deliver a

reasonable value premium; this is what we refer to as the “cash-flow risk puzzle.” Although

our results depend on the specifics of our model, we believe this is a general result pertaining

to external habit formation models a la Campbell and Cochrane (1999), as these models tend

to generate a variation in the stochastic discount factor that induces a growth premium on

stocks whenever there are no differences in cash-flow risk. Indeed, within a different model,

Lettau and Wachter (2007) found a similar result, which they resolve by assuming an unpriced

“sentiment” factor that drives the market price of risk. Here, we take a different route. We

maintain the habit formation specification, but assume instead that stocks differ in their cash-

flow risk, which generates several novel implications, such as that value stocks, endogenously,

have a larger cash-flow risk than growth stocks, and that the CAPM fails because of the implied

time variation in expected consumption growth. Next section highlights additional empirical

predictions of the model for the cross-section of stock returns.

24

6 Understanding asset pricing tests

As seen in the previous section, our model is flexible enough to replicate the standard moments

of interest both in the time-series and the cross-section of stock returns, although at the expense

of a large dispersion of cash-flow risk across firms. Notwithstanding this drawback, which is the

analog for the cross-section of the high correlation between returns and consumption shocks in

the original Campbell and Cochrane (1999) model (see, e.g., their Table 7 and the discussion

therein), we can build on this ability of the model to produce plausible magnitudes in the

cross-section to shed light on the economics behind many of the asset pricing tests that have

been proposed in the literature, as well as obtain new testable predictions. First, our model

features a time varying value premium, a novel implication of our framework, and we start

by showing evidence that this is indeed the case in the data and that our model matches to

a surprising degree this time-series variation. We then turn to some standard asset pricing

tests and reinterpret them in light of our model. Thus, for instance, we use our model to

construct an HML like factor, as in Fama and French (1993), and show the reasons for its

good performance as a successful predictor in the cross-section. Similarly, we revisit some

of the conditional CAPM tests that have been proposed recently in the literature. In our

model, the CAPM does not hold either conditionally or unconditionally, but we show that

(misspecified) conditional CAPM tests can “look better” than their unconditional counterpart

precisely because they capture the time-series variation of the value premium that the model

produces as well. In these additional exercises, we use the parameters φ = 0.07, ν = 0.55 and

θCF = 0.345%, which is, as discussed, our benchmark case.

6.1 The dynamics of the value premium

A novel implication of our framework is the fact that the value premium fluctuates as a

result of the interaction of the two key ingredients of the model, the strong discount effects of

habit persistence models with the cross-sectional dispersion of cash-flow risk as measured by

θCF > 0. To gauge the presence of this time-series variation in the data, Table 5 Panel A shows

the average excess return of the first and tenth decile portfolio, as a function of whether the

market-to-book ratio of the market portfolio is above or below a certain percentile, denoted by

c, for the 1948-2001 sample. Thus, the first line shows that the average excess rate of return

of the first decile (growth) portfolio is 13.18% if the market-to-book of the market portfolio

is below the 15th percentile of its empirical distribution and that of the tenth decile (value)

portfolio is 23.57%. The value premium is then 10.38%. Instead, when the market-to-book is

25

above the 15th percentile, the first decile portfolio has an average excess return of 5.73% and

the tenth portfolio has one of 10.35% for a total value premium of 4.62%, which is considerably

lower than the previous one. This pattern holds for any cut-off point: The value premium is

higher whenever the market-to-book of the market portfolio is low, which are also periods

where the average excess return of the market is high, as shown in the columns headed by RM .

Panel B of Table 5 reports the same calculations as in Panel A, but in simulated data.

The only difference is that, naturally, instead of using the market-to-book, we use the price-

dividend ratio of the market portfolio to identify the state. The pattern is indeed very similar

with the only exception of the level of the premiums which is, as already discussed, lower

than in the data. The value premium is higher when the price-dividend ratio of the market

portfolio is low than when it is high. For instance, when the price-dividend ratio of the market

portfolio is below the 15th percentile, the value premium is 10.90% whereas when it is above,

it is only 4.15%, very close to their empirical counterparts. In summary then, the discount risk

effects needed to replicate the time-series properties of the market portfolio interact with the

cross-sectional dispersion in cash-flow risk to generate variation in the value premium: Value

stocks are particularly risky during bad times, periods when the aggregate market premium

and its dividend yield are high relative to their unconditional mean, both in the data and the

model.25

6.2 The CAPM and the Fama-MacBeth regressions

As discussed at the beginning of Section 5, the CAPM is not able to price stocks sorted

by market-to-book, the value premium puzzle. We already discussed the value premium puzzle

in Section 5.2 and the performance of our model then; thus, for brevity, we do not repeat the

comments here. However, there is one noteworthy point to make in relation to the standard

tests of the CAPM via Fama and MacBeth (1973) cross-sectional regressions. To illustrate

the issue, the first line of Panel A of Table 7 reports the classic results about the failure of

the CAPM via Fama-MacBeth regressions: the intercept is positive, the market premium is

negative, and the cross-sectional R2 is small. Line 5 of Panel B in the same table reports the

performance of Fama-MacBeth cross-sectional regressions in artificial data. As can be seen,25To further investigate the properties of the conditional variation in the value premium, we also ran a

regression of the return on HML on the value spread, the difference between the dividend yield of the value

and growth portfolio, as in Cohen, Polk, and Vuolteenaho (2003, Table V), and find that the value spread is a

statistically significant predictor of the return on HML for several horizons. In the interest of space, we do not

report these results, which are available from the authors upon request.

26

the CAPM produces a good fit with an R2 of 91% and thus, it appears that in our model, the

unconditional CAPM works well, in contrast with our earlier findings in Table 3. The reason

for this difference is that in our simulated data, the CAPM betas correlate positively with

average excess returns (see Table 3 and Fig. 5). Thus, cross-sectional regressions that impose

no constraints on the level of estimated market premium tend to induce a good fit as measured

by the R2. But this is misleading, as the rejection of the model comes from the comparison of

the market premium implied by the cross-sectional regression, which is 10.24% (= 2.56%× 4),

and the market premium computed from the time-series of returns, which is 4.35% (see Table

3 Panel A.) That is, in general, the cross-sectional R2 is a poor indicator of the performance

of an asset pricing model.26

6.3 Understanding HML and the Fama and French (1993) model

In their seminal paper, Fama and French (1993) advance a new factor, HML, that is

able to correctly price value-sorted portfolios. Since then, the Fama and French (1993) model

has become a standard benchmark in asset pricing tests. How well does an HML factor work

in our setup? To answer this question, we construct an HML factor in artificial data that is

long the three top decile portfolios and short the bottom three. Table 6 presents the results of

time-series regressions,

Rpt = α + βMRM

t + βHMLRHMLt + εp

t for p = 1, 2, · · · , 10.

Panel A shows the results in the case of the empirical data. The results are well-known.

The intercepts go down considerably and only one of them is statistically significant; value

(growth) stocks have a large (small) loading on HML and the inclusion of HML in the time-

series regression collapses the betas on the market portfolio around 1.0 (see Fama and French,

1993, pp. 21–26).

Panel B shows the time-series regression in simulated data. Turning first to the loadings

on the market portfolio, notice that, as it was the case in the empirical sample, adding HML

to the time-series regressions has the effect of reducing the spread in the estimates of βiM and

collapse them around 1.0. As Fama and French (1993) note, this pattern is related to the

negative correlation between the market and the returns on HML.26This message has recently been emphasized by Lewellen and Nagel (2006) and Daniel and Titman (2006):

A small but slightly positive cross-sectional covariation between betas and average returns can result in the

unwarranted support of asset pricing models that fail to impose economically based restrictions on the size of

the premiums of the proposed factors.

27

As for the loading on the HML portfolio, notice that it has a strong cross-sectional

variation which reflects the cross-sectional variation in the underlying cash-flow risk of the

different portfolios. Indeed, the loading on HML of the growth portfolio is −0.28, whereas that

of the value portfolio is 1.07. Also the size of the intercepts of the time-series regressions drop

considerably relative to the size of the intercepts when only the market portfolio is present.27

Moreover, there is no longer any pattern in the variation of the intercept across decile portfolios,

which shows that HML is capturing the systematic pattern of misspricing shown in Panel A.

The evidence in the Fama-MacBeth regression confirms the time-series evidence. Line 2

of Table 7 Panel A shows that HML enters significantly and the estimated size of the premium

on HML is very close to the average excess return of the HML portfolio. This is also the case

in our simulated regression, which is shown in line 6 of Panel B in Table 7. The coefficient on

the loading on HML is very similar to its empirical counterpart and, once annualized, close to

our estimated average excess return on the HML portfolio, which is 3.21%. Thus, the inclusion

of HML in the cross-sectional regression aligns the portfolios correctly, as the intercept is now

close to zero and the (quarterly) market premium equals 1.31%, which annualized is 5.24%,

still higher than the average market return in simulation (4.35%), but much smaller than the

one obtained for the CAPM case.

6.4 Conditional asset pricing models

Conditional asset pricing models have been proposed recently to address the inability

of the CAPM to explain the value premium. The idea, as advanced by Hansen and Richard

(1987), is that the CAPM may fail unconditionally, but may hold conditionally, and thus, tests

of the CAPM that ignore conditioning information are misspecified. Researchers have reacted

to this observation by using as proxies for investors’ information set variables that are known

to forecast returns in the time-series.28 Typically, this has led to tests of multifactor models

where the additional factor, other than the market, is the market itself interacted with the

proposed conditioning variables.

Lines 3 and 4 in Panel A of Table 7 show that conditioning by the dividend yield of

the market portfolio and the cay variable of Lettau and Ludvigson (2001) results also in a27Notice that the value-weighted sum of the alphas should be equal to zero. Given that the only negative

alpha is that of the growth portfolio, it must be the case that some of the assets in the growth portfolio must

have extreme prices. We thank Gene Fama for pointing this out to us.28See, among others, the conditional asset pricing models of Jagannathan and Wang (1996), Ferson and

Harvey (1999), Lettau and Ludvigson (2001), and Santos and Veronesi (2006).

28

coefficient for the instrumented market that is strongly significant. In addition, the R2 is

an impressive 83% and 81%, respectively. Line 7 of Panel B shows that our model does

well also in this dimension. When we interact the returns of the market portfolio with the

simulated dividend yield of the market portfolio, we obtain a coefficient of similar magnitude

to its empirical counterpart.29 These conditional asset pricing models capture the fact that

value stocks become relatively riskier in bad times, a feature for which our model provides an

explanation, as shown in Table 5 and Fig. 6. In our setup, the conditional CAPM does not

hold, but it does better than its unconditional counterpart because it captures the conditional

effects that arise out of the interaction of discount effects with the cross-sectional dispersion

in θiCF .

7 Conclusions

Two sources of risk combine to determine the time-series properties of the market portfolio and

the cross-sectional properties of stock returns: discount risk and cash-flow risk. Campbell and

Cochrane (1999) argue that time variation of the market price of risk−i.e., discount risk−is

important to reconcile many empirical facts about the aggregate market portfolio. We show

that this channel though imposes tight restrictions on the cash-flow properties of value versus

growth stocks. Specifically, the natural growth premium that habit formation preferences

generate on firms that only differ in their expected dividend growth requires a large cross-

sectional dispersion in cash-flow risk across firms to generate a value premium.

Under this restriction, the model performs well, yielding most of the stylized facts about

the time-series and the cross-section of stock returns. In particular, besides matching the

conditional properties of the market portfolio, as in Campbell and Cochrane (1999), our model

also generates a sizable value premium, a countercyclical value premium, the failure of the

CAPM, and the better performance of factor models and conditional CAPM models.

Yet, in order to match the properties of both the aggregate market portfolio and the cross-

section of stock returns, the large dispersion of cash-flow risk that we must assume generates

a “cash-flow risk puzzle”: In our simulations, as in the data, value stocks have (endogenously)

higher cash-flow risk than growth stocks, but too much dispersion in cash-flow risk is required

to generate the value premium observed in the data. The cash-flow risk puzzle may arise from

various sources, such as the much larger noise in the cash-flow data compared to our simulated29We do not report the results for cay as in our setting, cay is perfectly correlated with log(D/P ).

29

data, which would tend to decrease the size of the “cash-flow beta” due to attenuation bias.

We view our model as a first step into understanding the sources of risk that explain both the

time-series and the cross-section of stock returns. Indeed, an important message of this paper

is that we cannot study one set of empirical facts independently of the other: any story that

attempts to quantitatively explain the cross-section of stock returns must also be consistent

with the time-series properties of the market portfolios. Otherwise, the parameterization that

is used to obtain quantitative predictions at the cross-sectional level may be quite misleading.

30

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Appendix A. Some additional resultsThe Aggregation Problem: It is useful to see the nature of the difficulty of imposing a general equilibrium

restriction Ct =Pn

j=1 Djt when working with multiple assets. To understand these restrictions and the nature

of our Assumptions 1 and 2, define Dt =`D1

t , ..., Dnt

´′and assume that

dDit

Dit

= µiD (Dt) dt + ν′idBt (27)

for some generic drifts µiD (Dt). Assume that νi is an n × 1 constant vector, and dBt is a n × 1 vector of

Brownian motions. From the general equilibrium restriction Ct =Pn

j=1 Djt and Ito’s Lemma, the process for

aggregate consumption isdCt

Ct= µc (st) dt + σc (st)

′ dBt, (28)

where st =`s1

t , ..., snt

´′=`D1

t /Ct, , ..., Dnt /Ct

´are shares of consumption produced by dividends, and

µc (st) =

nXi=1

sitµ

iD and σc (st) =

nXi=1

sitνi. (29)

The main difficulty in obtaining tractable expressions for asset prices lies in the dependence of µc (st) and

σc (st) on the shares st. Assumptions 1 and 2 in the body of the paper restrict the volatility of consumption

to a constant, but retain the time variation in the drift rate of consumption, although in a specific form. Our

specific assumptions not only allow us to obtain closed-form solutions for assets, but the mild predictability in

expected consumption growth also invalidates both the conditional and unconditional CAPM.

34

We note that in this setting, the cash-flow risk is given by

Cov

„dDi

t

Dit

,dCt

Ct

«= ν′iσc (st) =

nXj=1

sjtν′iνj . (30)

Finally, an application of Ito’s Lemma to sit = Di

t/Ct when the processes of Dit and Ct are given by (27)

and (28) shows that the volatility of the share process dsit is as in expression (5).

The habit dynamics: If Xt = λR t

−∞ eλ(τ−t)Cτdτ , we have dXt = λ (Ct −Xt) dt. Define then Gt =

f (Ct, Xt) = (Ct/ (Ct −Xt))γ . We then have

fC = −γGt

„G

1γt − 1

«C−1

t

fCC =

(γ (γ − 1) G

„G

1γt − 1

«2

+ 2γ

„G

1γt − 1

«G

1γ

+1

t

)C−2

t

fX = γGt1

(Ct −Xt),

where we used G1γt = Ct/ (Ct −Xt) and G

1γt − 1 = Xt/ (Ct −Xt). Ito’s Lemma then yields

dGt =˘µG (Gt)− σG (Gt) µc,1 (st)

¯dt− σG (Gt) σcdB1

t ,

where

µG (Gt) = γλGt +1

2γ (γ − 1) G

„G

1γt − 1

«2

σ2c + γ

„G

1γt − 1

«G

1γ

+1

t σ2c − σG (Gt) µc (31)

σG (Gt) = γGt

„G

1γt − 1

«. (32)

Appendix B. Proof of propositionsOur strategy to obtain prices and returns in our economy is standard. Given (13) , the stochastic discount

factor is given by

mt = e−ρt (Ct −Xt)−γ = e−ρtC−γ

t Gt.

We use Ito’s Lemma and our assumptions on the dynamics of Ct and Gt = S−γt to obtain

dmt

mt= −rf

t dt + σ′mdBt,

where the first, and only non-zero, entry in the diffusion component vector, σm, is given by

σ1m = − [γ + α (1− λSγ

t )] σc. (33)

Then we exploit our assumptions on the dynamics of Ct, Gt = S−γt , and si

t to solve for

P it = Et

»Z ∞

t

„mτ

mt

«Di

τdτ

–= Et

»Z ∞

t

„mτ

mt

«si

τCτdτ

–(34)

in closed-form. We then use (34) to compute returns and calculate the expected excess returns

Et

hdRi

t

i= −cov

„dmt

mt, dRi

«= −σ′mσi

R, (35)

35

where σiR is the diffusion component associated with the returns of asset i.

Proof of Proposition 1. This is a corollary to Proposition 2, and it is proved below.

Proof of Proposition 2. The pricing formula is

P it = Et

»Z ∞

t

e−ρ(τ−t) uc (Cτ , Xτ )

uc (Ct, Xt)Di

τdτ

–

= Cγt G−1

t Et

»Z ∞

t

e−ρ(τ−t)C1−γτ Gτsi

τdτ

–.

We divide the proof in two parts: First, we obtain a general pricing formula which depends on the state variables.

Second, we obtain analytical solutions for the coefficients of these state variables.

Part a.1: A pricing formula. For this proof, it is convenient to rewrite the share processes in its general form as

dsit =

nXj=1

sjtλjidt + si

t

“νi−s′tν

”dBt,

where λji = φsi, for i 6= j, and λii = −Pj 6=i λij = −φP

j 6=i sj = −φ`1− si

´= φsi − φ. Define the two

quantities

qit = C1−γ

t Gtsit and pi

t = C1−γt si

t

and the 2n× 1 vector yt = [qt,pt]. An application of Ito’s Lemma and tedious algebra shows

dyt = bΛyytdt + Σy,tdBt,

where

bΛy =

"Λ′ + bΘq

bΘqp

0 Λ′ + bΘp

#,

Λ =φ (s× 1′n) , bΘi for i = q, p, qp are diagonal matrices with ii element given by

bθi

q = (1− γ) µc −1

2γ (1− γ) σ2

c − k − (1− γ) σ2cα + (1− γ) θi − αθi

bθi

qp = kG + (1− γ) σ2cαλ + αλθi

bθi

p = (1− γ) µc −1

2γ (1− γ) σ2

c + (1− γ) θi,

and Σy,t is an appropriate matrix. Assuming existence of the expectation in the pricing function, we can apply

Fubini’s theorem

P it = Cγ

t G−1t Et

»Z ∞

t

e−ρ(τ−t)yiτdτ

–= Cγ

t G−1t

Z ∞

t

Et

he−ρ(τ−t)yi

τ

idτ.

The expectation in the integral can be computed as follows: Let ω be the vector of eigenvalues of bΛy,heω(τ−t)

i

the diagonal matrix with ii element given by eωi(τ−t), and U the matrix of associated eigenvectors. Then, we

can write

Et

he−ρ(τ−t)yi

τ

i= ιi ·U·

heω(τ−t)

i·U−1 · yte

−ρ(τ−t) =

2nX

k=1

2nXj=1

uike(ωk−ρ)(τ−t) ˆu−1jk

˜yjt,

wherehu−1

jk

iis the jk element of U−1. Substituting into the expectation, and taking the integral, we find

Z ∞

t

Et

he−ρ(τ−t)yi

τ

idτ =

2nX

k=1

2nXj=1

uik

hu−1

jk

i

ρ− ωkyjt =

2nXj=1

bijyjt,

36

where

bij =

2nX

k=1

uik

hu−1

kj

i

ρ− ωk.

Below, we obtain these coefficients in closed-form. Note, however, that by substituting yjt = qjt for j = 1, ..., n

and yjt = pj−n,t for j = n + 1, .., n we obtain

P it = Cγ

t G−1t Et

»Z ∞

t

e−ρ(τ−t)yiτdτ

–= Cγ

t G−1t

nX

j=1

bi1jqj,t +

nXj=1

bi2jpj,t

!

= Cγt G−1

t

C1−γ

t Gt

nXj=1

bi1js

jt + C1−γ

t

nXj=1

bi2js

jt

!

= Ct

nXj=1

“bi1j + bi

2jSγt

”sj

t .

Part a.2: Analytical formulas for bi1,j and bi

2,j . We finally obtain a closed form formula for bij ’s, and thus, of

bi1j and bi

2j . First, note that we can write

bij = ιi ·U·

`Ω−1´ ·U−1ιj ,

where Ω is the matrix with the eigenvalues of Iρ−bΛy on the principal diagonal. But then, since U· `Ω−1´·U−1 =“

Iρ− bΛy

”−1

, we have that for i = 1, ..., n and j = 1, ..., 2n

bij = ιi ·

“Iρ− bΛy

”−1

· ιj .

We now explicitly compute these quantities. Define B =“Iρ− bΛy

”−1

, so that

B“Iρ−bΛy

”= I.

Making this explicit, for every i = 1, .., n (row) we have

2nXj=1

bij

“Iρ−bΛy

”j

= ιi,

where“Iρ−bΛy

”j

is the jth row of“Iρ−bΛy

”and ιi is a (1× 2n) row vector with 1 in ith position, and zero

elsewhere. For every i, we have a system of equations that pins down bij for all j = 1, .., 2n. We now solve

this system of equations. To limit the number of indices involved, we do this exercise for i = 1. Of course, the

methodology works for every i. For i = 1 we have then the following two systems of equations. The first holds

for j = 1, .., n and the second for the remaining n rows:

b11

“ρ− φs1 + φ− bθ1

q

”−

nXj=2

b1jφsj = 1 (row 1)

−b11φs1 + b1

2

“ρ− φs2 + φ− bθ2

q

”−

nXj=3

b1jφsj = 0 (row 2)

...

−n−1Xj=1

b1jφsj + b1

n

“ρ− φsn + φ− bθn

q

”= 0 (row n)

37

−b11θ

1qp + b1

n+1

“ρ− φs1 + φ− bθ1

p

”−

nXj=2

b1n+jφsj = 0 (row n + 1)

−b12θ

2qp − b1

n+1φs1 + b1n+2

“ρ− φs2 + φ− bθ2

p

”−

nXj=3

b1n+jφsj = 0 (row n + 2)

...

−b1nθn

qp −n−1Xj=1

b1n+jφsj + b1

2n

“ρ− φsn + φ− bθn

p

”= 0 (row 2n).

The first set of equations is readily solved. In fact, we can write

b11 = α1

q + α1q × φ

nXj=1

b1js

j

b1k = αk

q × φ

nXj=1

b1js

j for k = 2, .., n,

where

αiq =

1“ρ + φ− bθi

q

” .

Multiply both sides of each row k = 1, ..., n by sk and sum across rows to obtain

nXj=1

b1js

j = s1α1q +

nXj=1

sjtα

jq

φ

nXj=1

b1js

j

!.

Define the constants

Hq =

nXj=1

sjαjq and Kq =

1

1− φHq.

Solving forPn

j=1 b1js

j we obtain the quantity

nXj=1

b1js

j = s1α1qKq.

Thus,

b11 = α1

q + α1q × φs1α1

qKq (36)

b1k = αk

q × φs1α1qKq for k = 2, .., n. (37)

Hence, the first term in the price-consumption ratio obtained earlier, i.e.,

P 1t

Ct=

nXj=1

b11js

jt +

nXj=1

b12js

jtS

γt ,

is given bynX

j=1

b11js

jt = α1

qs1t + φs1α1

qKq

nXj=1

αkqsj

t ,

where recall that for j = 1, ..., n we defined earlier b11j = b1

j .

We now turn to the second system of equations, which for k = 1, ..., n can be rewritten as

b1n+k = αk

pφ

nXj=1

b1n+js

j + b1kαk

pbθk

qp

38

with

αkp =

1“ρ + φ− bθk

p

”

and b1k given in (36) - (37). Substitute b1

k first, to obtain

b1n+1 = α1

pφ

nXj=1

b1n+js

j + α1pq + α1

pq × φs1α1qKq

b1n+k = αk

pφ

nXj=1

b1n+js

j + αkpqφs1α1

qKq,

where

αkpq = αk

pbθk

qpαkq .

As before, for k = 1, .., n multiply both sides by sk and sum across k’s to obtain

nX

k=1

skb1n+k = α1

pqs1 +

nX

k=1

skαkp

!φ

nXj=1

b1n+js

j +

nX

k=1

skαkpq

!φs1α1

qKq.

Let

Hp =

nX

k=1

skαkp

!,

and solve forPn

k=1 skb1n+k to find

nX

k=1

skb1n+k = α1

pqs1Kp +

nX

k=1

skαkpq

!φs1α1

qKqKp,

where

Kp =1

(1− φHp).

Substitute back into b1n+1 and b1

n+k and find

b1n+1 = α1

pq + s1g11

b1n+k = s1g1

k,

where for k = 1, ..., n

g1k = α1

qφ

(αk

p

α1

pθ1pqKp +

nX

j=1

sjαjpq

!φKqKp

!+ αk

pqKq

).

Thus, the second part in the price-consumption ratio is given by

nXj=1

b12js

jt = α1

pqs1t + s1

nX

k=1

g1ksk

t .

Generalizing the above derivations for every i = 1, ...,n, we can finally write

P it

Dit

= αi0 + αi

1Sγt + αi

2 (st)

„si

sit

«+ αi

3 (st)

„si

sit

«Sγ

t ,

39

where

αi0 = αi

q =1“

ρ + φ− bθi

q

”

αi1 = αi

pq =bθi

pq“ρ + φ− bθi

q

”“ρ + φ− bθi

p

”

αi2 (st) = φαi

qKq

`s′t αq

´

αi3 (st) = s′t gi,

where

gik = αi

qφn

αkp

“αi

pbθi

pqKp +`s′ αpq

´φKqKp

”+ αk

pqKq

o

and

bθi

q = (1− γ) µc − (1− γ)

„1

2γ + α

«σ2

c − k + (1− γ − α) θi

bθi

qp = kG + (1− γ) σ2cαλ + αλθi

bθi

p = (1− γ) µc −1

2γ (1− γ) σ2

c + (1− γ) θi.

Proof of Proposition 3. The diffusion component of stock returns is given by

σiR,t =

Sγt α (1− λSγ

t )

f i1

“si

sit, st

”+ Sγ

t

σc +

0@ 1

1 + f i2 (St, st)

“si

sit

” + ηiit

1Aσi

D (st) +X

j 6=i

ηijtσ

jD (st) . (38)

In fact, we can write

P it = Ct

“αi

0sit + αi

2 (st) si +“αi

1sit + αi

3 (st) si”

Sγt

”.

Define by eSt = Sγt = G−1

t . Using Ito’s Lemma, it is immediate to see that the diffusion of deS is given by

σS (Sγ) = Sγt α (1− λSγ

t ) σc.

Thus, an application of Ito’s Lemma shows that the diffusion term of P it is given by

σiR,t = σc +

`αi

1sit + αi

3 (st) si´Sγ

t α (1− λSγt )`

αi0s

it + αi

2 (st) si +`αi

1sit + αi

3 (st) si´Sγ

t

´σc

+

nX

k=1

( `αi

0 + αi1S

γt

´1i + φαi

qKqαkq + gi

k`αi

0sit + αi

2 (st) si +`αi

1sit + αi

3 (st) si´Sγ

t

´)

skt σk (st) ,

where 1i is the indicator function for k = i. Since σiD (st) = σc + σi (st), and since by construction

nX

k=1

( `αi

0 + αi1S

γt

´1k=i + φαi

qKqαkq + gi

k`αi

0sit + αi

2 (st) si +`αi

1sit + αi

3 (st) si´Sγ

t

´)

skt = 1,

we can rewrite

σiR,t =

Sγt α (1− λSγ

t )

f i1

`si/si

t; st

´+ Sγ

t

σc +

nX

k=1

( `αi

0 + αi1S

γt

´1k=1 + φαi

qKqαkq + gi

k`αi

0sit + αi

2 (st) si +`αi

1sit + αi

3 (st) si´Sγ

t

´)

skt σk

D (st)

=Sγ

t α (1− λSγt )

f i1

`si/si

t; st

´+ Sγ

t

σc +

8<:

1

1 + f2 (S; st)“

si

sit

” + ηii,t

9=;σi

D (st) +X

k 6=i

ηik,tσ

kD (st) ,

40

where

f i1

“si/si

t; st

”=

αi0 + αi

2 (st)`si/si

t

´

αi1 + αi

3 (st)`si/si

t

´

f2 (S; st) =αi

2 (st) + αi3 (st) Sγ

t

αi0 + αi

1Sγt

and

ηik,t =

`φαi

qKqαkq + gi

k

´sk

t`αi

0sit + αi

2 (st) si +`αi

1sit + αi

3 (st) si´Sγ

t

´ . (39)

Note also that

f ′1 < 0 if and only ifαi

2 (s)

αi3 (s)

<αi

0

αi1

=1

αipbθpq

.

Finally, the expected return is obtained from σR,t by using the formula

Et

hdRi

t

i= −Covt

„dRi

t,dm

mt

«.

Q.E.D.

Proof of Proposition 1. The price-consumption ratio of the total wealth portfolio can be obtained by simply

adding the prices of individual securities. In particular, we find

αTW0 (st) =

nXi=1

αiqs

it +

nXi=1

φsiαiqKq

nXj=1

αkqsj

t =`1 + φKqs

′ αq

´α′q st

αTW1 (st) =

nXi=1

αipqs

it +

nXi=1

φsinX

k=1

nαk

p

“αi

pqKp + sαpqφαiqKqKp

”+ αk

pqαiqKq

osk

t .

Algebra shows

αTW0 (st) =

1

1− φHqα′q st

αTW1 (st) =

1

1− φHq((αpst) Kpφsαpq + αpqst) .

We now turn to the computation of the volatility and expected returns of the total wealth (TW) portfolio.

An application of Ito’s Lemma to P TWt = Ct

`αTW

0 (st) + αTW1 (st) Sγ

t

´implies that the diffusion part of the

TW portfolios is given by

σTWP,t = σc +

αTW1 (st)

αTW0 (st) + Sγ

t × αTW1 (st)

Sγt α (1− λSγ

t ) σc

+αq + Sγ

t (Kpφsαpqαp + αpq)

α′q st + Sγt × ((αpst) Kpφsαpq + αpqst)

I (st) σ (st)

= σc +Sγ

t α (1− λSγt )

fTW1 (st) + Sγ

t

σc +

nXj=1

˘αj

q + Sγt

`Kpφsαpqα

jp + αj

pq

´¯sj

tPnk=1

˘αk

q + Sγt ×

`Kpφsαpqαk

p + αkpq

´¯sk

t

“νj − s′ · ν

”

=Sγ

t α (1− λSγt )

fTW1 (st) + Sγ

t

σc +

nXj=1

wTWjt σD (st)

with

fTW1 (st) =

αTW0 (st)

αTW1 (st)

,

41

and where

wTWjt =

˘αj

q + Sγt

`Kpφsαpqα

jp + αj

pq

´¯sj

tPnk=1

˘αk

q + Sγt ×

`Kpφsαpqαk

p + αkpq

´¯sk

t

are weights such thatP

j wTWjt = 1. Given the form of the stochastic discount factor, we obtain

Et

hdRTW

t

i= −Covt

„dRTW

t ,dmt

mt

«

= (γ + α (1− λSγ))

(Sγ

t α (1− λSγt )

fTW1 (st) + Sγ

t

σc +

nXj=1

wTWjt σjc

CF,t

).

Q.E.D.

Appendix C. Simulation results under different parameterizationsIn this appendix we discuss the simulations results under different parameterizations, contained in Tables

A.1 and A.2.

C.1 Return characteristics under different parameterizations

Columns 1 to 3 of Table A.1 report the combination of cash-flow parameters used in simulations. Columns

4 to 11 report the results for the total wealth portfolio and risk-free rate, while columns 12 and 13 report the

value premium. In boldface we report the benchmark case discussed in the previous sections.

The properties of the total wealth portfolio return (columns 4 and 5), interest rate (columns 6 and 7)

and return predictability (columns 8 to 11) are empirically reasonable across parameter choices, although some

differences exist across the various cash-flow parameter combinations. These differences are driven by general

equilibrium restrictions, as the properties of the aggregate portfolio depend upon the properties of individual

stock returns, and thus, on the characteristics of the individual cash-flow dynamics. The main impact of the

general equilibrium restrictions on the total wealth portfolio is that it induces a small predictable component in

expected consumption growth (see the discussion in Section 5.2.) In particular, as we increase the cash-flow risk

θCF , the term µc,1 (st) = s′t θCF in (3), which governs the forecastable component in expected consumption

growth, varies more and as a result, consumption growth becomes more predictable. This higher predictability

of consumption growth decreases the average equity premium. To reiterate the intuition, our model features

a low elasticity of intertemporal substitution and this translates into a declining price/consumption ratio in

the presence of an increase in expected consumption growth; because the latter is positively correlated with

consumption shocks, it follows that the equity premium declines as θCF increases. This argument is also behind

the lower long-return predictability in our model relative to other external habit persistence models.

Turning now to the cross-section of stock returns, column 12 of Table A.1 reports the difference in average

return between the value portfolio (portfolio 10, with low P/D ratio) and the growth portfolio (portfolio 1, with

high P/D ratio). We see that for each level of φ and for each level of share volatility ν, as we increase the

cash-flow risk parameter from θCF = 0 to θCF = 0.345%, the value premium moves from negative (i.e. a growth

premium) to positive. The intuition behind this result was discussed in Sections 4.2 and 4.3 and summarized

in Fig. 2 and 4. Table A.1 shows that indeed, the model is able to quantitatively match the value premium for

several parametric specifications, in addition to the one used in our benchmark case (in boldface).

Column 13 of Table 3 shows the CAPM fitted value premium. For each simulated portfolio i we compute

the CAPM implied expected return, ri = βirm, where rm is the simulated average market excess return, and

42

βi = cov(rit, r

mt )/var(rm

t ) is the market beta. The last column reports the difference in CAPM expected return

between the value portfolio (portfolio 10) and the growth portfolio (portfolio 1). If the CAPM gave a good

description of the model, we would obtain a spread identical to the difference in average return in column 12.

The last column of Table A.1 shows that this is not the case when θCF is high. Why does the CAPM fail in our

setting when θCF is high? In our model, a high θCF yields a mild time variation in expected consumption growth

through the term µc,1 (st) = s′t θCF , a time variation which, as discussed earlier in Section 5.2, invalidates the

CAPM.

At this point, it is perhaps useful to return to the evidence concerning the CAPM in the longer sample.

As we showed in Table 1 Panel C-2, and unlike the postwar sample, betas and average returns correlate positively

in the cross-section in the 1926-2001 sample and the CAPM can, at least partially, explain some of the value

premium. If generating a big gap between the simulated value premium and the CAPM fitted value premium

is not an objective, one can greatly improve the performance of our model, particularly in what refers to the

market portfolio. Indeed, consider the parameterization φ = 0.07, ν = 0.40 and θCF = 0.3%. In this case,

notice that as shown in Table A.1, the value premium is a respectable 4.91%, close to the empirical counterpart

in the long sample of 5.76%, but more importantly the simulated equity premium is a robust 7.06% and the

volatility is 17.37%, which are much closer to the empirically observed values (see Table 1). Thus, the model

is quite capable of matching the fundamental moments in the time-series and the cross-section at the expense

of a relatively good CAPM fit of the value premium, which is consistent with the evidence in the long sample.

But as we show next, the cash-flow risk puzzle obtains in all these alternative parameterizations.

C.2 Cash-flow characteristics under different parameterizations

Table A.2 shows the properties of the cash-flow processes under the different parameterizations. Our

purpose is to evaluate whether different parameterizations of the model imply cash-flow betas which are closer

to their empirical counterparts, as opposed to the benchmark case. columns 6 and 7 of Table A.2 report the

cash-flow beta for the growth portfolio (portfolio 1) and the value portfolio (portfolio 10), respectively, while

column 8 reports their spread. For each level of φ and each volatility ν, as we increase the cash-flow risk

dispersion parameter θCF , the cash-flow beta of the growth portfolio decreases, while the one of the value

portfolio increases, in line with the empirical evidence. As shown in Table A.1, in order for the model to yield a

value premium of 5% or more, we must have a dispersion of cash-flow risk θCF of at least 0.2%, independently

of the other cash-flow risk parameters, φ and ν. The last column of Table A.2 shows that for such values, the

implied cash-flow beta spread is too high.

An alternative way of seeing this “excessive” cash-flow risk that the model seems to require to undo the

strong discount effects implied by external habit persistence models, is to look at the implied dispersion in the

correlation coefficients between dividend and consumption growth across individual assets. The minimum and

maximum correlation in simulation is contained in columns 4 and 5 in Table A.2. Again, for those parameter

combinations in which a value premium arises, the dispersion of correlations is large: For instance, when

φ = 0.07, ν = 0.55, and θCF = 0.345%, we have that the dispersion of correlations between dividend growth

and consumption across assets is [ρ, ρ] = [−37.44%, 42.44%], which are rather large. This can be compared, for

instance, with the correlation between aggregate dividend and consumption growth in quarterly data which is

at most 0.2 and thus, the presence of substantial idiosyncratic risk should imply a lower correlation coefficient

for individual assets.

43

Table 1.

Basic moments in empirical data: 1948 – 2001.

Panel A: Summary statistics for the market portfolio. RM

and vol`RM

´the annualized mean and standard

deviation, respectively, of the excess returns of the market portfolio over the three-month Treasury bill. rf and

vol(rf ) are the mean and standard deviation, respectively, of the real risk-free rate, as measured by three-month

Treasury bill rate at the end of each quarter minus the expected (CPI) inflation computed from an AR(4).

Panel B: Predictability quarterly regressions of excess returns at the 1-, 2-, 3-, and 4-year horizons on the

log of the price-dividend ratio of the market portfolio. t-Stat denotes the Newey-West t-statistic where the

number of lags is the double of the forecasting horizon. Panel C-1: Summary statistics for the cross-section

of stock returns for the sample period 1948–2001. R is the annualized average excess returns of each of the

decile portfolios, ME/BE is the average market-to-book, and P/D the average price-dividend ratio. CAPM

β is obtained by running time-series regressions of excess return on each of the ten decile portfolios sorted on

ME/BE on the market excess return, where ME is the market equity and BE is the book value. CAPM α

denotes the intercepts of the time-series regression and the t(α) is the heteroskedasticity corrected t-statistic.

Quarterly dividends, returns, market equity, and other financial series are obtained from the CRSP-Compustat

database. The construction of the BE/ME sorted portfolios follows the standard procedure of Fama and French

(1992): Each year t portfolios are sorted into ten BE/ME-sorted portfolios using book-to-market ratios for year

t − 1. Returns on each of these portfolios are calculated from July of year t to June of year t + 1. Panel

C-2: Annualized monthly returns on value-sorted decile portfolios and their corresponding betas for the sample

period 1926:07–2001:12 from Ang and Chen (2007), Table 1 Panel A.

44

Panel A: Summary statistics for the market portfolio

RM

vol(RM ) rf vol(rf )

7.71% 16.25% 1.48% 2.15%

Panel B: Predictability regressions

Panel B-1: Sample 1948–2001

Horizon 4 8 12 16

ln`

DP

´0.13 0.20 0.26 0.35

t-stat. (2.13) (1.65) (1.34) (1.29)

R2 0.09 0.10 0.11 0.14

Panel B-2: Sample 1948 –1995

4 8 12 16

0.28 0.48 0.63 0.78

(4.04) (4.00) (4.49) (5.41)

0.19 0.32 0.43 0.54

Panel C-1: The value premium 1948 – 2001

Growth Value

Portf. 1 2 3 4 5 6 7 8 9 10

R (%) 6.86 7.77 7.67 7.63 8.53 9.96 8.39 11.00 11.39 12.36

ME/BE 5.05 2.68 2.00 1.63 1.38 1.18 1.01 0.86 0.70 0.45

P/D 43.47 31.38 26.87 24.65 22.65 21.62 20.64 19.95 20.00 21.77

Sharpe ratio 0.352 0.450 0.452 0.461 0.555 0.640 0.522 0.657 0.644 0.600

CAPM β 1.13 1.02 1.01 0.95 0.88 0.89 0.88 0.91 0.92 0.98

CAPM α −0.46 −0.03 −0.02 0.07 0.44 0.78 0.40 0.99 1.07 1.20

t(α) (−2.00) (−0.18) (−0.14) (0.32) (2.07) (3.73) (1.51) (3.73) (3.32) (2.65)

Panel C-2: The value premium 1926–2001

Growth Value

Portf. 1 2 3 4 5 6 7 8 9 10

R (%) 7.08 8.28 8.16 7.56 9.12 9.12 10.08 11.52 12.96 12.84

CAPM β 1.01 0.98 0.95 1.06 0.97 1.07 1.13 1.14 1.31 1.42

CAPM α −0.08 0.05 0.06 −0.06 0.11 0.06 0.09 0.21 0.21 0.14

t(α) (−1.16) (0.84) (1.08) (−0.92) (1.61) (0.72) (0.88) (1.90) (1.50) (0.76)

45

Table 2.

Model parameters used in the simulation.

Panel A: µc is the annual average growth rate of the consumption process, σc is the standard deviation of

consumption growth, γ is the coefficient controlling the local curvature of the utility function, ρ is the subjective

discount rate, G, λ, α, and k are the parameters controlling the dynamics of the process Gt = S−γt , where

St = (Ct −Xt)C−1t is the surplus-consumption ratio and the process for Gt is given by

dGt =ˆk`G−Gt

´− α (Gt − λ) µc,1 (st)˜dt− α (Gt − λ) σcdB1

t .

Panel B: The share process for i = 1, 2 · · · , n is

dsit = φ

“si − si

t

”+ si

tσi(st)dB

′t.

n = 200 is the number of assets in our artificial economy. θiCF is the parameter controlling the cash-flow risk.

Each asset is assigned a value of θiCF , which are distributed uniformly in the range above. si is the fraction

that each asset contributes to consumption in the steady state and φ is the speed of mean-reversion of the

share process. Finally, σi(st) = νi − s′tν where νi are vectors with νi,0 = θiCF /σc, νi,i =

qν2 − ν2

0,i, and the

remaining entries equal to zero. The simulation consists of 10,000 years of daily data.

Panel A: Consumption and preference parameters

µc σc γ ρ γ/S minγ/St α k

0.02 0.015 1.5 0.072 48 27.75 77 0.13

Panel B: Share process parameters

n θCF si φ ν

200 [0, 0.1%, 0.2%, 0.3%, 0.345%] 0.005 [0.05, 0.07, 0.1] [0.25, 0.4, 0.55]

46

Table 3.

Basic moments in simulated data.

Moments of interest in simulated data, which consist of 10,000 years of quarterly data for 200 firms. To construct

the portfolios, we sort on simulated price-dividend ratios following the standard procedures in the literature.

The parameters for the simulation are the ones reported in Table 2 with φ = 0.07, ν = 0.55, and θCF = 0.00345.

Panel A: Summary statistics for the market portfolio. RM

and vol`RM

´are the annualized mean and standard

deviation, respectively of the excess returns of the market portfolio over the three-month Treasury bill. rf is

the average risk-free rate and vol(rf ) is its annualized standard deviation. Panel B: Predictability quarterly

regressions of excess returns at the 1-, 2-, 3-, and 4-year horizons on the log of the price-dividend ratio of the

market portfolio. Panel C: Annualized average returns R, average log price-dividend ratio, ln (P/D), CAPM β,

and CAPM α. CAPM fitted returns are the returns resulting from multiplying the CAPM betas by the average

excess return of the market portfolio reported in Panel A.

Panel A: Summary statistics for the aggregate portfolio

RM

vol(RM ) rf vol(rf )

4.35% 13.03% 0.69% 4.36%

Panel B: Predictability regressions

Horizon 4 8 12 16

ln`

DP

´0.10 0.17 0.22 0.27

R2 0.03 0.04 0.05 0.06

Panel C: The value premium

Growth Value

Portf. 1 2 3 4 5 6 7 8 9 10

R (%) 3.07 3.58 4.37 4.77 5.27 5.45 5.84 6.00 6.43 8.23

ln (P/D) 6.38 5.07 4.613 4.35 4.12 3.90 3.68 3.44 3.15 2.68

Sharpe ratio 0.260 0.271 0.307 0.313 0.331 0.328 0.336 0.330 0.334 0.366

CAPM β 0.84 0.91 0.98 1.05 1.10 1.13 1.16 1.20 1.22 1.26

CAPM α −0.15 −0.09 0.02 0.06 0.12 0.13 0.20 0.20 0.29 0.68

CAPM fitt. ret. (%) 3.67 3.94 4.28 4.55 4.78 4.91 5.05 5.21 5.29 5.50

47

Table 4. Cash-flow betas.

Panel A: This panel reports the results of Cohen, Polk, and Vuolteenaho (2009, Table II, Panel B) for the

following regressions with annual data for each of the ten decile portfolios sorted on market-to-book,

4Xj=0

ρjROEpt+j,j+1 = βp

CF,0 + βpCF,1

4Xj=0

ρjROEMt+j + εp

4

4Xj=0

ρj Xpt+j,j+1

MEpt+j−1,j

= βpCF,0 + βp

CF,1

4Xj=0

ρj XMt+j,j+1

MEMt+j−1

+ εp4

P4j=0 ρjXp

t+j,j+1

MEpt−1,0

= βpCF,0 + βp

CF,1

P4j=0 ρjXM

t+j,j+1

MEMt−1

+ εp4

Xpt+4,j+4 −Xp

t−1,0

MEpt−1,0

= βpCF,0 + βp

CF,1

„XM

t+4 −XMt−1

MEMt−1

«+ εp

4

4Xj=0

ρj∆dpt+j,j+1 = βp

CF,0 + βpCF,1

4Xj=0

ρj∆dMt+j + εp

4.

ROE denotes the ratio of clean-surplus earning (Xt = BEt − BEt−1 + Dt where BEt−1 is the beginning-of-

the-period book equity and Dt are the dividends from CRSP) to BEt−1. MEt−1 denotes the market value at

the beginning of the period and ∆dpt+j,j+1 is the log of dividend growth of decile portfolio p. The first subscript

refers to the year of observation and the second to the number of years after the portfolio formation in the

sorting procedure. Similar quantities are defined for the market portfolio. GMM standard errors computed

using the Newey-West formula with four lags and leads are reported in parentheses. ρ is a constant, linked

to one minus the dividend yield, set at 0.95. Panel B: The first line reports the regression in simulated data

which corresponds to the fifth regressionP4

j=0 ρj∆dpt+j,j+1 above. The second line, Avge(θi

CF ) × 100, reports

the average cash-flow parameter for each of the ten decile portfolios.

48

Panel A: Empirical data from Cohen, Polk, and Vuolteenaho (2009)

Cash-flow definition Growth Value

1 2 3 4 5 6 7 8 9 10 10-1P4

j=0 ρjROEpt+j,j+1 0.72 0.91 0.94 0.96 0.95 0.96 0.97 1.11 1.28 1.51 0.79

std. err. (0.50) (0.31) (0.12) (0.25) (0.14) (0.12) (0.14) (0.24) (0.32) (0.30) (0.21)

P4j=0 ρj X

pt+j,j+1

MEpt+j−1,j

0.35 0.66 0.93 1.18 1.25 1.61 1.91 2.92 4.09 11.05 10.70

std. err. (0.32) (0.31) (0.17) (0.17) (0.28) (0.68) (1.00) (2.19) (3.23) (10.58) (4.17)

P4j=0 ρjX

pt+j,j+1

MEpt−1,0

0.48 0.72 0.96 1.13 1.22 1.40 1.50 2.00 3.13 7.63 7.14

std. err. (0.18) (0.18) (0.17) (0.15) (0.19) (0.35) (0.41) (1.18) (2.21) (8.85) (3.45)

Xpt+4,j+4−X

pt−1,0

MEpt−1,0

0.21 0.66 1.46 1.61 0.24 1.83 2.74 5.50 2.38 2.64 2.43

std. err. (0.19) (0.08) (0.52) (0.28) (0.61) (0.60) (1.24) (2.69) (0.60) (1.65) (0.57)

P4j=0 ρj∆dp

t+j,j+1 0.79 0.90 0.96 1.03 1.32 1.42 1.12 1.44 1.37 1.20 0.41

std. err. (0.19) (0.14) (0.10) (0.13) (0.27) (0.45) (0.30) (0.91) (0.74) (0.92) (0.41)

Panel B: Simulated data

Growth Value

1 2 3 4 5 6 7 8 9 10 1-10P4

j=0 ρj∆dpt+j,j+1 −7.40 −4.84 −2.47 −1.04 -0.05 0.75 1.44 2.05 2.64 4.73 12.13

Avge(θiCF )× 100 −0.2858 −0.1589 −0.0665 −0.0083 0.0295 0.0568 0.0787 0.0958 0.1128 0.1431 .4289

49

Table 5.

The dynamics of the value premium.

Panel A: Annualized average excess returns in the 1948–2001 sample for the growth (portfolio 1) and value

(portfolio 10) portfolios depending on whether the market-to-book of the market portfolio is below or above

the c percentile of its empirical distribution. Panel B: Annualized average excess returns in simulated data of

the growth (portfolio 1) and value (portfolio 10) portfolios depending on whether the simulated price-dividend

ratio of the market portfolio is below or above the c percentile of its distribution in simulated data. RM

is the

average excess return on the market portfolio in empirical data (Panel A) and simulated data (Panel B).

Panel A: Annualized average excess returns (%) in empirical data

Market-to-book of market portfolio < c

c 1 10 10-1 RM

15% 13.18 23.57 10.38 15.40

20% 10.57 21.70 11.14 13.41

25% 5.51 19.16 13.64 9.89

30% 6.97 19.49 12.51 10.50

35% 8.19 18.65 10.45 11.14

Market-to-book of market portfolio > c

c 1 10 10-1 RM

15% 5.73 10.35 4.62 6.34

20% 5.95 10.06 4.11 6.31

25% 7.31 10.11 2.80 6.99

30% 6.82 9.32 2.50 6.62

35% 6.15 8.98 2.83 5.87

Panel B: Annualized average excess returns (%) in simulated data

Price-dividend of market portfolio < c

c 1 10 10-1 RM

15% 7.37 18.27 10.90 10.43

20% 6.56 16.07 9.51 9.22

25% 5.96 14.60 8.64 8.36

30% 5.50 13.46 7.96 7.67

35% 5.13 12.60 7.47 7.18

Price-dividend of market portfolio > c

c 1 10 10-1 RM

15% 2.30 6.46 4.15 3.27

20% 2.19 6.26 4.07 3.13

25% 2.10 6.10 4.00 3.01

30% 2.02 5.98 3.96 2.92

35% 1.95 5.87 3.92 2.82

50

Table 6.

The Fama-French (1993) model: Time series regressions (quarterly).

This table reports the results of time-series regressions

Rpt = α + βMRM

t + βHMLRHMLt + εp

t for p = 1, 2, · · · , 10

in empirical (Panel A) and simulated (Panel B) data of returns on each of the book-to-market sorted portfolios on

the market excess return and the returns on HML, where βHML is the regression coefficient on HML. Quarterly

dividends, returns, market equity, and other financial series are obtained from the CRSP-Compustat database

for the period 1948 – 2001. The construction of the BE/ME sorted portfolios follows the standard procedure

of Fama and French (1992): Each year t portfolios are sorted into ten BE/ME-sorted portfolios using book-to-

market ratios for year t − 1. Returns on each of these portfolios are calculated from July of year t to June of

year t + 1. The HML portfolio is constructed by taking long and short position in the top and bottom three

decile portfolios, respectively. For Panel B, the simulations consist of 10,000 years of quarterly artificial data.

Returns on portfolios and HML are computed using the same procedure as in the data, except we sort stocks

by their price-dividend ratios rather than their book-to-market ratios, which are not available in simulations.

Panel A: Empirical data

Growth Value

Portf. 1 2 3 4 5 6 7 8 9 10

α 0.20 0.17 0.02 −0.12 0.19 0.28 −0.40 0.01 −0.08 −0.36

t(α) (1.13) (1.05) (0.14) (−0.61) (0.87) (1.58) (−2.15) (0.09) (−0.43) (−1.23)

βM 1.04 0.99 1.00 0.98 0.91 0.96 0.99 1.05 1.09 1.20

t`βM´

(43.68) (51.25) (46.13) (35.28) (30.25) (38.66) (39.90) (48.04) (39.61) (29.85)

βHML −0.42 −0.12 −0.03 0.12 0.16 0.31 0.50 0.61 0.72 0.97

t`βHML

´(−12.13) (−2.37) (−0.68) (1.88) (3.62) (8.85) (10.35) (15.52) (21.04) (14.14)

Panel B: Simulated data

Growth Value

Portf. 1 2 3 4 5 6 7 8 9 10

α −0.01 0.02 0.07 0.06 0.09 0.10 0.11 0.03 0.07 0.13

βM 0.93 0.97 1.01 1.05 1.08 1.11 1.11 1.10 1.09 0.93

βHML −0.28 −0.21 −0.09 −0.01 0.06 0.08 0.16 0.31 0.41 1.07

51

Table 7.

Asset pricing models: Fama-MacBeth regressions (quarterly).

Panel A: Fama-MacBeth regressions in empirical data for the sample 1948 – 2001. Line 1, CAPM regressions

where Mkt. represents the average excess return of the market portfolio. Line 2, Fama and French (1993) model,

where SMB is the return on “small minus big” and HML is the return on “high minus low.” Line 3, conditional

CAPM regression where the dividend yield, log(D/P), of the market portfolio is used as a conditioning variable.

Line 4 conditional CAPM regression where the variable cay of Lettau and Ludvigson (2001) is used as a

conditioning variable. Panel B: Fama-MacBeth regressions in simulated data. t-Statistics are in parentheses

and Adj. R2 is the adjusted R2.

Panel A: Empirical data

Const. Mkt. SMB HML Mkt×log(D/P) Mkt×cay Adj. R2

1. 4.69 −2.52 11%

(3.21) (−1.65)

2. 0.36 1.63 −0.31 1.05 80%

(0.23) (0.99) (−0.31) (2.16)

3. 2.72 −0.87 1.71 83%

(2.24) (−0.65) (2.46)

4. 3.06 −1.37 0.06 81%

(2.48) (−1.01) (2.34)

Panel B: Simulated data

Const. Mkt. HML Mkt×log(D/P) Adj. R2

5. −1.45 2.56 91%

6. −0.17 1.31 0.94 99%

7. 0.63 0.38 1.16 98%

52

Table A.1.

The market portfolio and the value premium in simulations: Robustness.

This table reports basic moments of the returns for three different values of ν, which determines the maximum

volatility of share process across assets, and the measure of cash-flow risk, θCF ≥ 0, which determines the support

on which the cash-flow risk parameters of individual firms are uniformly distributed, θiCF ∈ [−θCF , θCF ]. R

M

and vol`RM

´are the annualized mean and stadard deviation, respectively, of the excess returns of the market

portfolio over the three-month Treasury bill. rf is the average risk-free rate and vol(rf ) is its annualized standard

deviation. All these numbers are in percentages. b12 and b16 are the regression coefficients of the quarterly

predictability regressions of excess returns on the log of the price-dividend ratio of the market portfolio for the

three- and four-year horizons. R212 and R2

16 are the corresponding R2s. 10 − 1 denotes the value premium, in

percentages, defined as the difference between the average return on the value portfolio, portfolio 10, and the

growth portfolio, portfolio 1. CAPM 10− 1 is the fitted CAPM value premium, where the betas are calculated

the standard way in simulated data and the market premium is the corresponding RM

in each line. The numbers

in bold correspond to the benchmark case presented in more detail in Table 2.

53

Cash-flow risk Market portfolio Predictability Value premium

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13)

φ ν θCF × 100 RM

vol(RM ) rf vol`rf´

b12 R212 b16 R2

16 10− 1 CAPM 10− 1

0.05 0.25 0.00 9.90 24.16 1.16 5.44 0.64 24.69 0.74 28.40 -2.54 -2.59

0.05 0.25 0.30 6.32 15.57 0.78 4.48 0.43 10.82 0.50 12.22 8.59 8.48

0.05 0.40 0.00 9.90 24.16 1.16 5.44 0.64 24.69 0.74 28.40 -3.28 -3.38

0.05 0.40 0.30 6.40 15.97 0.79 4.52 0.43 11.03 0.49 12.57 6.61 5.84

0.05 0.55 0.00 9.90 24.16 1.16 5.44 0.64 24.69 0.74 28.40 -3.71 -3.82

0.05 0.55 0.30 6.61 17.03 0.81 4.71 0.36 9.98 0.42 11.35 4.53 2.81

0.07 0.25 0.00 9.90 24.16 1.16 5.44 0.64 24.69 0.74 28.40 -2.44 -2.53

0.07 0.25 0.10 9.69 23.62 1.12 5.32 0.63 23.92 0.73 27.51 -1.27 -1.42

0.07 0.25 0.20 8.95 21.79 1.00 4.97 0.61 21.25 0.70 24.36 2.40 2.34

0.07 0.25 0.30 7.02 17.22 0.79 4.46 0.53 14.14 0.60 15.89 7.51 7.45

0.07 0.25 0.35 3.97 10.23 0.67 4.20 0.22 4.23 0.27 4.98 7.10 6.70

0.07 0.40 0.00 9.90 24.16 1.16 5.44 0.64 24.29 0.74 28.40 -3.22 -3.37

0.07 0.40 0.10 9.69 23.63 1.12 5.33 0.63 23.87 0.73 27.45 -2.56 -2.77

0.07 0.40 0.20 8.96 21.83 1.00 4.99 0.61 21.16 0.70 24.26 -0.07 -0.25

0.07 0.40 0.30 7.06 17.37 0.80 4.49 0.52 14.17 0.60 15.95 4.91 4.62

0.07 0.40 0.35 4.09 11.14 0.68 4.23 0.25 5.63 0.30 6.69 6.29 4.57

0.07 0.55 0.00 9.90 24.16 1.16 5.44 0.64 24.69 0.74 28.40 -3.67 -3.86

0.07 0.55 0.10 9.70 23.66 1.13 5.34 0.63 23.19 0.73 27.34 -3.27 -3.48

0.07 0.55 0.20 8.99 21.95 1.01 5.05 0.60 20.92 0.69 23.95 -1.49 -1.70

0.07 0.55 0.30 7.15 17.85 0.81 4.60 0.49 13.66 0.57 15.43 2.83 2.19

0.07 0.55 0.35 4.35 13.03 0.69 4.36 0.22 5.39 0.27 6.48 5.16 1.83

0.10 0.25 0.00 9.90 24.16 1.16 5.44 0.64 24.69 0.74 28.40 -2.21 -2.34

0.10 0.25 0.35 6.23 15.43 0.70 4.21 0.48 11.05 0.55 12.17 7.38 7.29

0.10 0.40 0.00 9.90 24.16 1.16 5.44 0.64 24.69 0.74 28.40 -3.04 -3.23

0.10 0.40 0.35 6.26 15.56 0.70 4.23 0.48 11.23 0.55 12.41 5.30 4.97

0.10 0.55 0.00 9.90 24.16 1.16 5.44 0.64 24.69 0.74 28.40 -3.51 -3.80

0.10 0.55 0.35 6.33 15.93 0.70 4.30 0.47 11.29 0.54 12.51 3.57 2.89

54

Table A.2.

The properties of the cash-flow process in simulations: Robustness.

For each value of ν and θCF the table reports several moments of the cash-flow process in simulated data.

[ρ, ρ] stands for the range of the correlation coefficients between individual dividend growth and consumption

growth. β1CF,1 and β10

CF,1 correspond to the regression coefficients of the time-series regression

4Xj=0

ρj∆dpt+j,j+1 = βp

CF,0 + βpCF,1

4Xj=0

ρj∆dMt+j + εp

4,

as in the legend to Table 4 in simulated data for the Growth (p = 1) and Value (p = 10) portfolios. The

benchmark parameterization discussed in the text appears in boldface.

55

φ ν θCF × 100 [ ρ ρ ] β1CF,1 β10

CF,1 10-1

(1) (2) (3) (4) (5) (6) (7) (8)

0.05 0.25 0.00 4.61 7.17 1.06 0.97 -0.009

0.05 0.25 0.30 -76.14 80.65 -8.69 6.89 15.58

0.05 0.40 0.00 2.39 4.89 1.13 0.93 -0.20

0.05 0.40 0.30 -45.59 51.54 -7.33 5.38 12.71

0.05 0.55 0.00 1.41 3.83 1.27 0.98 -0.29

0.05 0.55 0.30 -31.57 36.52 -5.68 4.55 10.23

0.07 0.25 0.00 4.62 7.17 1.04 0.96 -0.08

0.07 0.25 0.10 -20.59 32.18 0.04 1.89 1.85

0.07 0.25 0.20 -47.93 57.09 -3.30 4.15 7.45

0.07 0.25 0.30 -76.25 80.83 -8.14 6.70 14.84

0.07 0.25 0.35 -88.89 91.05 -9.62 7.94 17.56

0.07 0.40 0.00 2.37 4.89 1.09 0.96 -0.13

0.07 0.40 0.10 -13.11 20.28 0.43 1.49 1.06

0.07 0.40 0.20 -29.05 36.23 -1.80 3.10 4.90

0.07 0.40 0.30 -45.78 51.91 -6.37 5.22 11.59

0.07 0.40 0.35 -53.42 58.85 -8.63 5.73 14.36

0.07 0.55 0.00 1.39 3.84 1.17 0.99 -0.18

0.07 0.55 0.10 -9.54 14.60 0.69 1.28 0.59

0.07 0.55 0.20 -20.60 25.98 -1.01 2.40 3.41

0.07 0.55 0.30 -32.10 37.34 -4.79 4.28 9.07

0.07 0.55 0.35 -37.44 42.44 -7.40 4.73 12.13

0.10 0.25 0.00 4.63 7.18 1.05 0.96 -0.09

0.10 0.25 0.35 -89.06 91.24 -8.94 7.75 16.69

0.10 0.40 0.00 2.37 4.91 1.08 0.98 -0.10

0.10 0.40 0.35 -53.47 59.03 -7.43 5.73 13.16

0.10 0.55 0.00 1.37 3.85 1.04 1.00 -0.04

0.10 0.55 0.35 -37.78 43.02 -5.92 4.76 10.68

56